aa r X i v : . [ m a t h . G R ] A p r OPERAD GROUPS AND THEIR FINITENESS PROPERTIES
WERNER THUMANN
Abstract.
We propose a new unifying framework for Thompson-like groupsusing a well-known device called operads and category theory as language.We discuss examples of operad groups which have appeared in the literaturebefore. As a first application, we proof a theorem which implies that planar orsymmetric or braided operads with transformations satisfying some finitenessconditions yield operad groups of type F ∞ . This unifies and extends existingproofs that certain Thompson-like groups are of type F ∞ . Contents
1. Introduction 21.1. Structure of the article 31.2. Notation and conventions 41.3. Acknowledgements 42. Preliminaries on categories 42.1. Comma categories 42.2. The classifying space of a category 52.3. The fundamental groupoid of a category 52.4. Coverings of categories 62.5. Contractibility and homotopy equivalences 72.6. Smashing isomorphisms in categories 82.7. Calculus of fractions and cancellation properties 92.8. Monoidal categories 112.9. Cones and joins 132.10. The Morse method for categories 143. Operad groups 183.1. Basic definitions 183.2. Normal forms 223.3. Calculus of fractions and cancellation properties 233.4. Operads with transformations 243.5. Examples 264. A topological finiteness result 324.1. Three types of arc complexes 334.2. A contractible complex 394.3. Isotropy groups 394.4. Finite type filtration 424.5. Connectivity of the filtration 434.6. Applications 53References 54
Mathematics Subject Classification.
Primary 20F65; Secondary 57M07, 20F05, 18D50.
Key words and phrases.
Thompson groups, operads, finiteness properties. Introduction
In unpublished notes of 1965, Richard Thompson defined three interesting groups
F, T, V . For example, F is the group of all orientation preserving piecewise linearhomeomorphisms of the unit interval with breakpoints lying in the dyadic rationalsand with slopes being powers of 2. It has the presentation F = (cid:10) x , x , x , . . . | x − k x n x k = x n +1 for k < n (cid:11) In the subsequent years until the present days, hundreds of papers have been de-voted to these and to related groups. The reason for this is that they have the abilityto unite seemingly incompatible properties. For example, Thompson showed that V is an infinite finitely-presented simple group which contains every finite groupas a subgroup. Even more is true: Brown showed in [7] that V is of type F ∞ which means that there is a classifying space for V with finitely many cells in everydimension. For F , this was proven by Brown and Geoghegan in [8]. They alsoshowed that H k ( F, Z F ) = 0 for every k ≥
0. This implies in particular that allhomotopy groups of F at infinity vanish and that F has infinite cohomological di-mension. Thus, they found the first example of an infinite dimensional torsion-freegroup of type F ∞ . In [6], Brin and Squier showed that F is a free group free group,i.e. contains no non-abelian free subgroups. Geoghegan conjectured in 1979 that F is non-amenable. If this is true, F would be an elegant counterexample to the vonNeumann conjecture. Ol’shanskii disproved the von Neumann conjecture around1980 by giving a different counterexample (see [32] and the references therein). De-spite several attempts of various authors, the amenability question for F still seemsto be open at the time of writing. During the 1970s, Thompson’s group F wasrediscovered twice: In the context of homotopy theory by Freyd and Heller [21] andin connection with a problem in shape theory by Dydak [12].Since the introduction of the classical Thompson groups F, T and V , a lot ofgeneralizations have appeared in the literature which have a “Thompson-esque”feeling to them. Among them are the so-called diagram or picture groups [25],various groups of piecewise linear homeomorphisms of the unit interval [39], groupsacting on ultrametric spaces via local similarities [26], higher dimensional Thomp-son groups nV [4] and the braided Thompson group BV [5]. A recurrent themein the study of these groups are topological finiteness properties, most notablyproperty F ∞ . The proof of this property is very similar in each case, going backto a method of Brown, the Brown criterion [7], and a technique of Bestvina andBrady, the discrete Morse Lemma for affine complexes [2]. This program has beenconducted in all the above mentioned classes of groups: For diagram or picturegroups in [14, 15], for the piecewise linear homeomorphisms in [39], for local simi-larity groups in [16], for the higher dimensional Thompson groups in [20] and forthe braided Thompson group in [9].The main motivation to define the class of operad groups, which are the centralobjects in this article, was to find a framework in which a lot of the Thompson-like groups could be recovered and in which the established techniques could beperformed to show property F ∞ , thus unifying and extending existing proofs in theliterature. The main device to define these groups are discrete operads. Operads arewell established objects whose importance in mathematics and physics has steadilyincreased during the last decades. Representations of operads constitute algebrasof various types and consequently find applications in such diverse areas as Lie-Theory, Noncommutative Geometry, Algebraic Topology, Differential Geometry,Field Theories and many more. To apply our F ∞ theorem to a given Thompson-like group, one has to find the operadic structure underlying the group. Then one PERAD GROUPS AND THEIR FINITENESS PROPERTIES 3 has to check whether this operad satisfies certain finiteness conditions. In a lot ofcases, the proofs of these conditions are either trivial or straightforward.1.1.
Structure of the article.
Our language will be strongly category theoryflavoured. Although we assume the basics of category theory, we collect and recallin Section 2 all the tools we will need for the definition of operad groups and forour main result. We lay a particular emphasis on topological aspects of categoriesby considering categories as topological objects via the nerve functor. This can bemade precise by endowing the category of (small) categories with a model structureQuillen equivalent to the usual homotopy category of spaces, but we won’t use thisfact. In Subsection 2.10, we will discuss a tool which is probably not so well-knownas the others. There, we introduce the discrete Morse method for categories inanalogy to the one for simplicial complexes: With the help of a Morse function,a category can be filtered by a nested sequence of full subcategories. The relativeconnectivity of such a filtration is controlled by the connectivity of certain categoriesassociated to each filtration step, the so-called descending links. This can be usedto compute lower bounds for the connectivity of categories.In Section 3, we will introduce the main objects of this article, the so-calledoperad groups. Before we do this, we recall the notion of operads (internal to thecategory of sets). This is an abstract algebraic structure generalizing that of amonoid. It comes with an associative multiplication and with identity elements.However, elements in an operad, which are called operations, can be of higher arity(or degree): An operation posseses several inputs and one output. If we have anoperation with n inputs, then we can plug the outputs of n other operations intothe inputs of the first one, yielding composition maps for the operad. This conceptcan be generalized even more: Just as one proceeds from monoids to categories byintroducing further objects, we can introduce colors to operads and label the inputsand outputs of operations with these colors. Then we require that the compositionmaps respect this coloring. Furthermore, we can introduce actions of the symmetricor braid groups on the inputs of the operations and obtain symmetric or braidedoperads.We then attach, in a very natural way, a category to each operad, called thecategory of operators. When taking fundamental groups of these categories, wearrive at the concept of operad groups. In Subsection 3.5, we then discuss someexamples of operads and corresponding operad groups. We will see that all of theThompson-like groups mentioned in the first part of the introduction can be realizedas operad groups. Furthermore, we give new examples and even a procedure howto generate a lot of these Thompson-like groups as operad groups associated tosuboperads of endomorphism operads.We will also discuss so-called operads with transformations which are operadswith invertible degree 1 operations. In this context, we will introduce very elemen-tary and elementary operations. These model in some sense the generators andrelations in such an operad with transformations. In particular, we can define whatit means for such an operad to be finitely generated or of finite type. This will beimportant in Section 4 where we prove the following Theorem.
Let O be a finite type (symmetric/braided) operad with transformationswhich is color-tame and such that there are only finitely many colors and degree operations. Assume further that O satisfies the cancellative calculus of fractions.Then the operad groups associated to O are of type F ∞ . The conditions are explained in the text and are usually not hard to verify inpractice. The proof proceeds roughly as follows and the ideas are mainly inspiredby [2, 7, 9, 20, 39]. Denote by S the category of operators of O . We then can WERNER THUMANN look at the universal covering category U of S which is contractible due to theconditions in the theorem. We mod out the isomorphisms in U and obtain thequotient category U / G which is still contractible. The operad group Γ, which isthe fundamental group of S , acts on U by deck transformations. This induces anaction on U / G . Brown’s criterion applied to this action yields that Γ is of type F ∞ if we show that the isotropy groups of the action are of type F ∞ and if we find afiltration by invariant finity type subcategories with relative connectivity tendingto infinity. The latter is shown by appealing to the discrete Morse method forcategories mentioned earlier. Thus, we have to inspect the connectivity of certaindescending links. This is the hardest part of the proof. We filter each descendinglink by two subcategories, the core and the corona. The core is related to certainarc complexes in R d with d = 1 , ,
3. A lower bound for the connectivity of thesecomplexes is given in Theorem 4.9. The connectivity for the corona and for thewhole descending link is then deduced from the connectivity of the core by usingagain the discrete Morse method.1.2.
Notation and conventions.
When f : A → B and g : B → C are two com-posable arrows, we write f ∗ g or f g for the composite A → C instead of the usualnotation g ◦ f . Consequently, it is often better to plug in arguments from the left.When we do this, we use the notation x⊲f for the evaluation of f at x . However,we won’t entirely drop the usual notation f ( x ) and use both notations side by side.Objects of type Aut( X ) will be made into a group by the definition f · g := f ∗ g .Conversely, a group G is considered as a groupoid with one object and arrows theelements in G together with the composition f ∗ g := f · g .1.3. Acknowledgements.
I want to thank my adviser Roman Sauer for the op-portunity to pursue mathematics, for his guidance, encouragement and supportover the last few years. I also gratefully acknowledge financial support by the DFGgrants 1661/3-1 and 1661/3-2.2.
Preliminaries on categories
In this section, we review some aspects of category theory which we need forlater considerations. In particular, we want to emphasize the concept of seeingcategories as topological objects. Note that everything, except the Morse methodfor categories explained in Subsection 2.10, should be mathematical folklore andwe make no claim of originality.2.1.
Comma categories.
Let A f −→ C g ←− B be two functors. Then the commacategory f ↓ g has as objects all the triples ( A, B, γ ) where A resp. B is an objectin A resp. B and γ : f ( A ) → g ( B ) is an arrow in C . An arrow from ( A, B, γ ) to( A ′ , B ′ , γ ′ ) is a pair ( α, β ) of arrows α : A → A ′ in A and β : B → B ′ in B suchthat the diagram f ( A ) γ / / f ( α ) (cid:15) (cid:15) g ( B ) g ( β ) (cid:15) (cid:15) f ( A ′ ) γ ′ / / g ( B ′ )commutes. Composition is given by composing the components.If f is the inclusion of a subcategory, we write A↓ g for the comma category f ↓ g .Furthermore, if A is just a subcategory with one object A and its identity arrow, wewrite A ↓ g . In this case, the objects of the comma category are pairs ( B, γ ) where
PERAD GROUPS AND THEIR FINITENESS PROPERTIES 5 B is an object in B and γ : A → g ( B ) is an arrow. An arrow from ( B, γ ) to ( B ′ , γ ′ )is an arrow β : B → B ′ such that the triangle g ( B ) g ( β ) (cid:15) (cid:15) A γ < < ③③③③③③③③ γ ′ " " ❉❉❉❉❉❉❉❉ g ( B ′ )commutes. Of course, there are analogous abbreviations for the right factor.2.2. The classifying space of a category.
We assume that the reader is familiarwith the basics of simplicial sets (see e.g. [24]). The nerve N ( C ) of a category C isa simplicial set defined as follows: A k -simplex is a sequence A α −→ A α −→ . . . α k − −−−→ A k of k composable arrows. The i ’th face map d i : N ( C ) k → N ( C ) k − is given bycomposing the arrows at the object A i . When i is 0 or k , then the object A i is removed from the sequence instead. The i ’th degeneracy map s i : N ( C ) k → N ( C ) k +1 is given by inserting the identity at the object A i .The geometric realization | N ( C ) | of N ( C ) is a CW-complex which we call theclassifying space B ( C ) of C . See [43] for the reason why this is called a classifyingspace. If the category C is a group, then B ( C ) is the usual classifying space ofthe group which is defined as the unique space (up to homotopy equivalence) withfundamental group the given group and with higher homotopy groups vanishing.Since we can view any category as a space via the above construction, anytopological notion or concept can be transported to the world of categories. Forexample, if we say that the category C is connected, then we mean that B ( C ) isconnected. Of course, one can easily think of an intrinsic definition of connectednessfor categories and we will give some for other topological concepts below. But thereare also concepts for which a combinatorial description is at least unknown, forexample higher homotopy groups.Transporting topological concepts to the category CAT of (small) categories viathe nerve functor can be made precise: The Thomason model structure on
CAT [10, 40] is a model structure Quillen equivalent to the usual model structure on
SSET , the category of simplicial sets.Every simplicial complex is homeomorphic to the classifying space of some cat-egory: A simplicial complex can be seen as a partially ordered set of simpliceswith the order relation given by the face relation. Moreover, a partially orderedset (poset) is just a category with at most one arrow between any two objects.The classifying space of a poset coming from a simplicial complex is exactly thebarycentric subdivision of the simplicial complex.Even more is true: McDuff showed in [31] that for each connected simplicial com-plex there is a monoid (i.e. a category with only one object) with classifying spacehomotopy equivalent to the given complex. Thus, every path-connected space hasthe weak homotopy type of some monoid. For example, observe the monoid consist-ing of the identity element and elements x ij with multiplication rules x ij x kl = x il .In [17] it is shown that its classifying space is homotopy equivalent to the 2-sphere.2.3. The fundamental groupoid of a category.
Following the philosophy oftransporting topological concepts to categories via the nerve functor, we define thefundamental groupoid π ( C ) of a category C to be the fundamental groupoid of its WERNER THUMANN classifying space. There is also an intrinsic description of the fundamental groupoidof C in terms of the category itself which we will describe now (see e.g. [24, ChapterIII, Corollary 1.2] that these two notions are indeed the same up to equivalence).The objects of π ( C ) are the objects of C and the arrows of π ( C ) are paths modulohomotopy. Here, a path in C from an object A to an object B is a zig-zag ofmorphisms from A to B , i.e. starting from A , one travels from object to objectover the arrows of C , regardless of the direction of the arrows. For example, thefollowing zig-zag is a path in C A ← C → C ← C ← C → C → B Paths can be concatenated in the obvious way. The homotopy relation on paths isthe smallest equivalence relation respecting the operation of concatenation of pathsgenerated by the following elementary relations: A α −→ B β −→ C ∼ A αβ −−→ CA α ←− B β ←− C ∼ A βα ←−− CA α −→ B α ←− A ∼ AA α ←− B α −→ A ∼ AA id −→ A ∼ AA id ←− A ∼ A where the A ’s on the right represent the empty path at A . Composition in π ( C )is given by concatenating representatives. The identities are represented by theempty paths. If A is an object of C then we denote by π ( C , A ) the automorphismgroup of π ( C ) at A and call it the fundamental group of C at A .The fundamental groupoid of C has two further descriptions: First, denote by G the left adjoint functor to the inclusion functor from groupoids to categories.Then we have π ( C ) = G ( C ). Second, it is the localization C [ C − ] of C (at all itsmorphisms) since it comes with a canonical functor ϕ : C → π ( C ) satisfying thefollowing universal property: Having any other functor η : C → A with the propertythat η ( f ) is an isomorphism in A for every arrow f in C , then there is a uniquefunctor ǫ : π ( C ) → A such that ϕǫ = η . C ϕ / / η ! ! ❉❉❉❉❉❉❉❉❉ π ( C ) ǫ (cid:15) (cid:15) A Coverings of categories.
Let P : D → C be a functor. We say that P is acovering if for every arrow a in C and every object X in D which projects via P onto the domain or the codomain of a , there exists exactly one arrow b in D withdomain resp. codomain X and projecting onto a via P . In other words, arrowscan be lifted uniquely provided that the lift of the domain or codomain is given.Of course, P yields a map on the classifying spaces. To justify the definition ofcovering functor, we have the following: Proposition 2.1.
Let P : D → C be a functor. Then P is a covering functor ifand only if BP : B D → B C is a covering map of spaces.Proof. By [22, Appendix I, 3.2], BP = | N P | : | N D| → | N C| is a covering map if andonly if N P : N D → N C is a covering of simplicial sets as defined in [22, AppendixI, 2.1]. This means that every n -simplex in N C uniquely lifts to N D provided thatthe lift of a vertex of the simplex is given. The lifting property for P as definedabove says that this is true for 1-simplices. So it is clear that P is a covering functor PERAD GROUPS AND THEIR FINITENESS PROPERTIES 7 provided that BP is a covering map of spaces. For the converse implication, oneexploits special properties of nerves of categories. Not every simplicial set arisesas the nerve of a category. The Segal condition gives a necessary and sufficientcondition for a simplicial set to come from a category: Every horn Λ in for 0 < i < n can be uniquely filled by an n -simplex. Using this, the lifting property for 1-simplices implies the lifting property for n -simplices. (cid:3) Now let C be a category and X an object in C . Observe the canonical functor ϕ : C → π ( C ). Define U X ( C ) to be the category X ↓ ϕ . The canonical projection U X ( C ) → C sending an object ( B, γ ) to B is a covering. Furthermore, U X ( C ) issimply connected, i.e. connected and its fundamental groupoid is equivalent to theterminal category (see [33]). So it deserves the name universal covering category.More precisely, it is the universal covering of the component of C which containsthe object X .There is a canonical functor π ( C ) → CAT taking objects X to the category U X ( C ) and an arrow f : X → Y to a functor U X ( C ) → U Y ( C ) which is given byprecomposition with f − . Fixing the object X , this functor restricts to a functor π ( C , X ) → CAT sending the unique object of the group π ( C , X ) to the universalcovering U X ( C ). This is the same as a representation of π ( C , X ) in CAT , i.e. a grouphomomorphism ρ : π ( C , X ) → Aut (cid:0) U X ( C ) (cid:1) into the group of invertible functorswith multiplication given by f · g := f ∗ g . Equivalently, this is a right action of thegroup π ( C , X ) on U X ( C ) given by the formula α · γ := α⊲ ( γ⊲ρ ) for γ ∈ π ( C , X )and arrows α in U X ( C ). This gives the usual deck transformations on the universalcovering.2.5. Contractibility and homotopy equivalences.
We say that a category iscontractible if its classifying space is contractible and we say that a functor F : C →D is a homotopy equivalence if BF : B C → B D is one. There are some standardconditions which assure that a category is contractible or a functor is a homotopyequivalence. These will be recalled below.A non-empty category C is contractible ifi) C has an initial object.ii) C has binary products.iii) C is a generalized poset (see Definition 2.8) and there is an object X together with a functor F : C → C such that for each object X there existarrows X → F ( X ) ← X (compare with [35, Subsection 1.5]).iv) C is filtered which means that for every two objects X, Y there is an object Z with arrows X → Z , Y → Z and for every two arrows f, g : A → B thereis an arrow h : B → C such that f h = gh .Of course, the dual statements are also true. It is instructive to sketch the argu-ments for these four claims:i) Let I be the category with two objects and one non-identity arrow fromthe first to the second object. The classifying space of I is the unit interval I . A natural transformation of two functors f, g : C → D can be interpretedas a functor
C × I → D . On the level of spaces, this gives a homotopy B C × I → B D . If C is a category with initial object X , then there isa unique natural transformation from the functor const X (sending everyarrow of C to id X ) to the identity functor id C . On the level of spaces, thisyields a homotopy between id B C and the constant map B C → B C withvalue the point X .ii) Choose an object X in C . Let F : C → C be the functor Y X × Y .Projection onto the first factor yields a natural transformation F → const X and projection onto the second factor yields a natural transformation F → WERNER THUMANN id C . This gives two homotopies which together give the desired contractionof B C .iii) First note that, if F, G : C → C are two functors with the property thatthere is an arrow F ( X ) → G ( X ) for each object X , then this alreadydefines a natural transformation F → G by uniqueness of arrows in thegeneralized poset. Now the conditions on X and F yield that there arenatural transformations id C → F and const X → F . On the level of spacesthis gives the desired contraction of B C .iv) First, let D be a finite subcategory of C . We claim that there exists acocone over D in C , i.e. there is an object Z in C and for each object Y in D an arrow Y → Z which commute with the arrows in D . This cocone iscontractible because Z is a terminal object. A cocone can be constructedas follows: First pick two objects Y , Y in D and find an object Z ′ witharrows Y → Z ′ and Y → Z ′ . Pick another object Y and find an object Z ′′ with arrows Y → Z ′′ and Z ′ → Z ′′ . Repeating this with all objects of D , we obtain an object Q together with arrows f Y : Y → Q for every object Y in D . The f Y probably won’t commute with the arrows in D yet, but wecan repair this by repeatedly applying the second property of filteredness.Pick an arrow d : Y → Y ′ in D and observe the parallel arrows df Y ′ and f Y .Apply the second property to find an arrow ω : Q → Q ′ with df Y ′ ω = f Y ω .Replace Q by Q ′ and all the arrows f D for objects D in D by f D ω . Repeatthis with all the other arrows in D .Now to finish the proof of this item, take a map S n → B C . Since S n is compact, it can be homotoped to a map such that the image is coveredby the geometric realization of a finite subcategory. The cocone over thissubcategory then gives the desired null-homotopy.We recall Quillen’s famous Theorem A from [34] which gives a sufficient but ingeneral not necessary condition for a functor to be a homotopy equivalence. Theorem 2.2.
Let f : C → D be a functor. If for each object Y in D the category Y ↓ f is contractible, then the functor f is a homotopy equivalence. Similarly, ifthe category f ↓ Y is contractible for each object Y in D , then f is a homotopyequivalence.Remark . When applying this theorem to an inclusion f : A → B of a full subcategory, it suffices to check Y ↓ f = Y ↓A for objects Y not in A . If Y is anobject in A , the comma category Y ↓A has the object ( Y, id Y ) as initial object andthus is automatically contractible. Similar remarks apply to the comma categories f ↓ Y = A↓ Y . Remark . If D is a groupoid, then for Y, Y ′ ∈ D the comma categories Y ↓ f and Y ′ ↓ f are isomorphic. Thus one has to check contractibility only for one Y . Thesame remarks apply to the comma categories f ↓ Y .2.6. Smashing isomorphisms in categories.
Recall that a connected groupoidis equivalent, as a category, to any of its automorphism groups. Consequently,a connected groupoid is contractible if and only if its automorphism groups aretrivial. This is the case if and only if there is exactly one isomorphism between anytwo objects.Let C be a category and G ⊂ C a subcategory which is a disjoint union of con-tractible groupoids. We define the quotient category C / G as follows: The objects of C / G are equivalence classes of objects of C where we say that X ∼ Y are equivalentif there is an isomorphism X → Y in G . Note that such an isomorphism is uniquesince each component of G is contractible. We defineHom C / G (cid:0) [ X ] , [ Y ] (cid:1) := (cid:8) A → B in C (cid:12)(cid:12) A ∈ [ X ] , B ∈ [ Y ] (cid:9)(cid:14) ∼ PERAD GROUPS AND THEIR FINITENESS PROPERTIES 9 where two elements ( A → B ) ∼ ( A ′ → B ′ ) in the set are defined to be equivalentif the diagram A / / G∋ (cid:15) (cid:15) ✤✤✤ B ∈G (cid:15) (cid:15) ✤✤✤ A ′ / / B ′ commutes. Let [ α : A → B ] and [ β : C → D ] be two composable arrows, i.e. [ B ] =[ C ], then there is a unique isomorphism γ : B → C in G and one defines[ α : A → B ] ∗ [ β : C → D ] := [ αγβ : A → D ]Set id [ X ] = [id X ]. One easily checks that C / G is a well-defined category. Remark . Observe that if
X → Y is an arrow in C / G and representatives X and Y have been chosen for X and Y , then there is a unique arrow X → Y representing X → Y . Remark . Let X and Y be two objects in C / G . Fix some object X representing X . Then the arrows X → Y in C / G are in one to one correspondence with arrows X → Y in C modulo isomorphisms in G on the right. Likewise, if we fix some object Y representing Y , then arrows X → Y in C / G are in one to one correspondencewith arrows X → Y in C modulo isomorphisms in G on the left. Proposition 2.7.
The canonical projection p : C → C / G is a homotopy equivalence.Proof. We want to apply Quillen’s Theorem A (Theorem 2.2) to the projection p : C → C / G . Hence, we have to show that for each object [ X ] in C / G the commacategory [ X ] ↓ p is contracible. Indeed, it follows with Remark 2.5 that the object( X, id [ X ] ) in [ X ] ↓ p is an initial object. (cid:3) In the following, this technique will be applied primarily to generalized posets : Definition 2.8.
A generalized poset is a category such that α = β whenever α, β : A → B .Recall that a (honest) poset is a category with at most one arrow between anytwo objects (regardless of the direction of the arow). In a generalized poset C ,however, we allow objects to be uniquely isomorphic. Every subgroupoid G of ageneralized poset is a disjoint union of contractible ones and C / G is a generalizedposet again. If we collapse each connected component of the subgroupoid consistingof all the isomorphisms, we even get a homotopy equivalent (honest) poset whichwe call the underlying poset of the generalized poset.2.7. Calculus of fractions and cancellation properties.
The next definitionis very classical and due to Gabriel and Zisman [22].
Definition 2.9.
Let C be a category. It satisfies the calculus of fractions if thefollowing two conditions are satisfied: • ( Square filling ) For every pair of arrows f : B → A and g : C → A there arearrows a : D → B and b : D → C such that af = bg . D a / / b (cid:15) (cid:15) B f (cid:15) (cid:15) C g / / A • ( Equalization ) Whenever we have arrows f, g : A → B and a : B → C suchthat f a = ga , then there exists an arrow b : D → A with bf = bg . D b / / A g / / f / / B a / / C More precisely, this is called the right calculus of fractions. There is also a dual left calculus of fractions. Since we are mainly interested in the right calculus offractions, we omit the word “right”.
Remark . The existence of binary pullbacks in C trivially implies the squarefilling property but it also implies the equalization property [1, Lemma 1.2]. So acategory with binary pullbacks satisfies the calculus of fractions.The calculus of fractions has positive effects on the complexity of the fundamentalgroupoid π ( C ): One can show (see e.g. [22] or [3]) that each class in π ( C ) can berepresented by a span which is a zig-zag of the form • • o o / / • Furthermore, two spans • (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❅❅❅❅❅❅❅ • •• _ _ ❅❅❅❅❅❅❅ ? ? ⑧⑧⑧⑧⑧⑧⑧ are homotopic if and only if the diagram can be filled in the following way: • (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❅❅❅❅❅❅❅ • • o o / / O O (cid:15) (cid:15) •• _ _ ❅❅❅❅❅❅❅ ? ? ⑧⑧⑧⑧⑧⑧⑧ In other words, the elements in the localization can be described as fractions andthis explains the name of the calculus of fractions. We will frequently write ( α, β )for a span consisting of arrows α and β where the first arrow α points to the left(i.e. is the denominator) and the second arrow β points to the right (i.e. is thenominator). Two spans are composed by concatenating representatives to a zig-zagand then transforming the zig-zag into a span by choosing a square filling of themiddle cospan. • w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ (cid:127) (cid:127) (cid:31) (cid:31) ' ' ❖❖❖❖❖❖❖ • • o o / / • • o o / / • The canonical functor ϕ : C → π ( C ) is given by sending an arrow α to the classrepresented by the span • • id o o α / / • Using the special form of the homotopy relation from above, we see that two arrows α, β : X → Y are homotopic if and only if there is an arrow ω : A → X such that ωα = ωβ .We now turn to cancellation properties in categories. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 11
Definition 2.11.
Let C be a category. It is called right cancellative if f a = ga forarrows f, g, a implies f = g . It is called left cancellative if af = ag implies f = g .It is called cancellative if it is left and right cancellative. Remark . Note that we have the following implications:right cancellation = ⇒ equalizationequalization + left cancellation = ⇒ right cancellation Proposition 2.13.
Let C be a category satisfying the cancellative calculus of frac-tions. Then the canonical functor ϕ : C → π ( C ) is faithful and a homotopy equiv-alence.Proof. Injectivity is easy: Let f, g be arrows in C which are mapped to the samearrow in π ( C ). This means that f, g are homotopic. Since C satisfies the calculus offractions, this implies that there is an arrow ω with ωf = ωg . From the cancellationproperty it follows that f = g .For showing that the functor is a homotopy equivalence, we apply Quillen’sTheorem A (Theorem 2.2) to the functor ϕ : C → π ( C ). Let X be an objectin π ( C ), i.e. an object in C . We have to check that the comma category X ↓ ϕ iscontractible. Note that this is the universal covering category U X ( C ). First we claimthat this category is a generalized poset: Let ( Z, a ) and ( Z ′ , a ′ ) be objects in X ↓ ϕ and γ , γ be two arrows from ( Z, a ) to ( Z ′ , a ′ ). This means that Z, Z ′ are objectsin C , a : X → Z⊲ϕ and a ′ : X → Z ′ ⊲ϕ are arrows in π ( C ) and γ , γ : Z → Z ′ arearrows in C such that a ∗ ( γ ⊲ϕ ) = a ′ = a ∗ ( γ ⊲ϕ ) in π ( C ). It follows γ ⊲ϕ = γ ⊲ϕ and therefore γ = γ by injectivity.Now we want to show that this generalized poset is cofiltered. Then we canapply item iv) of Subsection 2.5. We have to show that for each two objects A, A ′ in X ↓ ϕ there is another object B and arrows B → A and B → A ′ . Let A = ( Z, a )and A ′ = ( Z ′ , a ′ ) with arrows a : X → Z⊲ϕ and a ′ : X → Z ′ ⊲ϕ which can berepresented by spans ( α, β ) and ( α ′ , β ′ ) respectively. Choose a square filling ( γ, γ ′ )of the cospan ( α, α ′ ). Z⊲ϕ • α ~ ~ ⑥⑥⑥⑥⑥ β < < ②②②②②② X Y O O ✤✤✤✤✤ (cid:15) (cid:15) ✤✤✤✤✤ o o ❴ ❴ ❴ ❴ ❴ γ c c ❋❋❋❋❋❋ γ ′ { { ①①①①①① • α ′ ` ` ❆❆❆❆❆ β ′ " " ❊❊❊❊❊❊ Z ′ ⊲ϕ Then the arrow ω := γα = γ ′ α ′ can be interpreted as the denominator of a spanrepresenting an arrow in π ( C ) which we denote by ω − . Furthermore, since ϕ : C → π ( C ) is the identity on objects, we can write Y = Y ⊲ϕ . Thus, we can define theobject B := ( Y, ω − ) in X ↓ ϕ . Finally, the arrows γβ and γ ′ β ′ give arrows B → A and B → A ′ respectively. (cid:3) Remark . The functor ϕ : C → π ( C ) in Proposition 2.13 is still a homotopyequivalence if we drop the cancellation property from the hypothesis. This is provedin [11, Section 7].2.8. Monoidal categories.
We assume that the reader is acquainted with the defi-nition of monoidal categories, symmetric monoidal categories and braided monoidalcategories (see e.g. [30]). In the following, we will always assume the strict versions, i.e. the associator, right and left unitor are identities. We frequently use the sym-bol I to denote the unit object. Moreover, for objects X and Y , the symbol γ X,Y denotes the natural braiding isomorphism X ⊗ Y → Y ⊗ X . We will sometimescall a monoidal category planar in order to stress that it’s neither symmetric norbraided.Joyal and Street introduced the notion of braided monoidal categories in [27].It is designed such that the braided monoidal category freely generated by a singleobject is the groupoid with components the braid groups B n . More precisely,we have an object for each natural number n , there are no morphisms n → m with n = m and Hom( n, n ) = B n . More generally, they indroduced the braidedmonoidal category Braid ( C ) freely generated by another category C [27, page 37]:The objects are free words in the objects of C , i.e. finite sequences of objects of C . A morphism consists of a braid β ∈ B n where the strands are labelled withmorphisms α i : A i → B i of C , yielding an arrow( β, α , . . . , α n ) : ( A ⊲β , . . . , A n⊲β ) → ( B , . . . , B n )in Braid ( C ). Composition is performed by composing the braids and applyingcomposition in C to every strand. The tensor product is given by juxtaposition,i.e. by ( β, α , . . . , α n ) ⊗ ( β ′ , α ′ , . . . , α ′ n ) = (cid:0) β ⊗ β ′ , α , . . . , α n , α ′ , . . . , α ′ n (cid:1) where β ⊗ β ′ means juxtaposition of braids. A set C can be viewed as a discretecategory, so we also obtain the notion of a braided monoidal category Braid ( C )freely generated by a set. The arrows are just braids with strands labelled by theelements of C , i.e. are colored.The same remarks apply to the symmetric version. In particular, a category C freely generates a symmetric monoidal category Sym ( C ).Even simpler, we can form the free monoidal category Mon ( C ) generated by acategory C . The strands are decorated by arrows in C but they are not allowed tobraid or cross each other.If C is a (symmetric/braided) monoidal category, then there is exactly one tensorstructure on π ( C ) making it into a (symmetric/braided) monoidal category andsuch that the canonical functor ϕ : C → π ( C ) respects that structure, i.e. ϕ ( X ⊗ Y ) = ϕ ( X ) ⊗ ϕ ( Y ) ϕ ( α ⊗ β ) = ϕ ( α ) ⊗ ϕ ( β ) ϕ ( I ) = Iϕ ( γ X,Y ) = γ ϕ ( X ) ,ϕ ( Y ) for objects X, Y and arrows α, β . The tensor product on the level of arrows can beconstructed as follows: Let one arrow be represented by the zig-zag A α ←− A α −→ · · · α k −−→ A k and the other arrow by the zig-zag B β −→ B β ←− · · · β l −→ B l Then the tensor product may be represented by the zig-zag A α ←− A α −→ · · · α k −−→ A k id −→ A k id ←− · · · id −→ A k ⊗ ⊗ ⊗ ⊗ · · · ⊗ ⊗ ⊗ ⊗ ⊗ · · · ⊗ ⊗ B ←− B −→ · · · id −→ B β −→ B β ←− · · · β l −→ B l PERAD GROUPS AND THEIR FINITENESS PROPERTIES 13
Cones and joins.
Let C , D be two categories. We define the join C ∗ D . Theset of objects of
C ∗ D is the disjoint union of the objects of C and D . The set ofarrows is the disjoint union of the arrows of C and D together with exactly one arrow C → D for each pair ( C, D ) of objects C of C and D of D . The composition rules arethe unique ones extending the compositions in C and D . The classifying space of thejoin is homotopy equivalent to the join of the classifying spaces B ( C ∗D ) ≃ B C ∗ B D .Now we define the cone over a category. The objects of Cone( C ) are the objectsof C plus another object called tip . The arrows are the arrows of C together withexactly one arrow from tip to every object in C . Dually, there is a Cocone( C ) over C . In the cocone, the extra arrows go from the objects of C to the extra object tip .Last but not least, when we have a join C ∗ D of two categories, there is a mixedversion Coone(
C ∗ D ) which we call the coone over the join. Again, there is oneextra object tip and for every object in
C ∗ D we have an extra arrow. When wehave an object in C , the extra arrow goes to tip . When we have an arrow in D ,the extra arrow comes from tip . The composition of an arrow C → tip with anarrow tip → D is the unique arrow C → D from the definition of the join. Allthree coning versions give the usual conings on the topological level: B (Cone( C )) ∼ = Cone( B ( C )) B (Cocone( C )) ∼ = Cone( B ( C )) B (Coone( C ∗ D )) ∼ = Cone( B ( C ∗ D ))The join of two spaces X and Y is defined to be the homotopy pushout of the twoprojections X ← X × Y → Y . Thus, it is defined only up to homotopy and thereis some freedom to choose models of a join. Indeed, there is another constructiongiving the join of two categories. For this, we need to recall the Grothendieckconstruction: Let J be some indexing category and F : J →
CAT a diagram in
CAT .The objects of the Grothendieck construction R F are pairs ( J, X ) of objects J in J and X in J⊲F . An arrow from (
J, X ) to ( J ′ , X ′ ) is a pair ( f, α ) consisting of anarrow f : J → J ′ and an arrow α : X⊲ ( f ⊲F ) → X ′ . Composition is given by( f, α ) ∗ ( f ′ , α ′ ) := (cid:0) f ∗ f ′ , α⊲ ( f ′ ⊲F ) ∗ α ′ (cid:1) In [40] it is shown that there is a model structure on
CAT
Quillen equivalent to
SSET ,nowadays called the Thomason model structure, and in [41] it is shown that thenerve of the Grothendieck construction R F is homotopy equivalent to the homotopypushout of the diagram F ∗ N which is obtained from the diagram F by applyingthe nerve functor. In fact, R F realizes the homotopy pushout of F with respect tothe Thomason model structure on CAT [19, Section 3].Now let C , D be categories. We call the Grothendieck construction of the diagram C pr C ←−− C × D pr D −−→ D the Grothendieck join of C and D and denote it by C ◦ D . From [41] we know that B ( C ◦ D ) is homotopy equivalent to the homotopy pushout of the diagram B C pr B C ←−−− B C × B D pr B D −−−→ B D But the latter is the join B C ∗ B D by definition. So we have B ( C ◦ D ) ≃ B C ∗ B D .One can show that the Grothendieck join is associative and thus we can write C ◦ . . . ◦ C k for a finite collection C i of categories. The objects of such an iteratedGrothendieck join are elements of the setobj( C ◦ . . . ◦ C k ) = a S ⊂{ ,...,k } Y s ∈ S obj( C s ) Whenever we have S ⊂ T ⊂ { , . . . , k } , objects ( Y t ) t ∈ T and ( X s ) s ∈ S and arrows α s : Y s → X s in C s for each s ∈ S , then there is an arrow( α s ) s ∈ S : ( Y t ) t ∈ T → ( X s ) s ∈ S For R ⊂ S ⊂ T the composition is given by( α s ) s ∈ S ∗ ( β r ) r ∈ R = ( α r ∗ β r ) r ∈ R There is also a dual notion of the Grothendieck join which we define as
C • D := ( C op ◦ D op ) op Since B ( A op ) = B ( A ) for any category, we still have B ( C • D ) ≃ B C ∗ B D . Further-more, it is still associative, so that we can write C • . . . • C k for a finite collection C i of categories. The objects of such an iterated dual Grothendieck join are elementsof the set obj( C • . . . • C k ) = a S ⊂{ ,...,k } Y s ∈ S obj( C s )Whenever we have S ⊂ T ⊂ { , . . . , k } , objects ( X s ) s ∈ S and ( Y t ) t ∈ T and arrows α s : X s → Y s in C s for each s ∈ S , then there is an arrow( α s ) s ∈ S : ( X s ) s ∈ S → ( Y t ) t ∈ T For R ⊂ S ⊂ T the composition is given by( β r ) r ∈ R ∗ ( α s ) s ∈ S = ( β r ∗ α r ) r ∈ R The Morse method for categories.
We first recall the Morse methodin the case of simplicial complexes which has been used in [2] to prove finitenessproperties of certain groups. We then explain the same method in the context ofcategories.Let C be a simplicial complex. Let v be a vertex in C . Denote by C − v the fullsubcomplex spanned by the vertices of C except v . Observe the link lk ( v ) of v in C which is contained in C − v . We then have a canonical pushout diagram lk ( v ) (cid:15) (cid:15) / / Cone (cid:0) lk ( v ) (cid:1) (cid:15) (cid:15) C − v / / C where Cone (cid:0) lk ( v ) (cid:1) denotes the simplicial cone over lk ( v ). The following lemmaexpresses the connectivity of the pair ( C , C − v ) in terms of the connectivity of lk ( v ). Lemma 2.15.
Let X and L be two spaces and L → C a cofibration into a con-tractible space C . Let L (cid:15) (cid:15) / / C (cid:15) (cid:15) X / / Z be a pushout of spaces. If L is ( n − -connected, then the pair ( Z, X ) is n -connected. The proof is a standard application of the Seifert–van Kampen theorem, theHurewicz theorem and the Mayer–Vietoris sequence for pushouts.More generally, let X be the full subcomplex of the simplicial complex X spanned by a subset of vertices. Then X can be built up from X by succes-sively adding vertices. We thus get a filtration X ⊂ X ⊂ . . . ⊂ X of X by fullsubcomplexes. If v is the vertex in X i which is not contained in X i − , then wedefine lk ↓ ( v ) := lk X i ( v ) = lk X ( v ) ∩ X i − to be the descending link of v . If all the descending links appearing this way arehighly connected, then also the pair ( X , X ) will be highly connected and, usingthe long exact homotopy sequence, we obtain the following: Proposition 2.16.
Let x ∈ X be a point. Assume that each descending link is n -connected. Then, we have π k ( X , x ) ∼ = π k ( X , x ) for k = 0 , . . . , n . Note that, in general, the descending links depend on the order in which thevertices are added. We call two vertices v , v in X \ X independent if they are notjoined by an edge in X . Assume now that we want to add v and then v at somestep of the process. The independence condition ensures that the descending linksof v and v do not depend on the order in which v and v are added.The adding order is often encoded in a Morse function. This is a function f assigning to each vertex in X \ X an element in a totally ordered set, e.g. N . Werequire that vertices with the same f -value are pairwise independent. We thenadd vertices in order of ascending f -values. Because of the independence prop-erty, the adding order of vertices with the same f -value can be chosen arbitrarily.Alternatively, we can add vertices with the same f -value all at once.We now give a version of this concept for categories. Let C be a category and X an object in C with Hom C ( X, X ) = { id X } . Define C − X to be the full subcategoryof C spanned by the objects of C except X . We define lk ↓ ( X ) := C − X ↓ X to be the descending up link of X and lk ↓ ( X ) := X ↓C − X to be the descending down link of X . Furthermore, define lk ↓ ( X ) := lk ↓ ( X ) ∗ lk ↓ ( X )to be the descending link of X .We have a commutative diagram D as follows: lk ↓ ( X ) (cid:15) (cid:15) / / Coone (cid:0) lk ↓ ( X ) ∗ lk ↓ ( X ) (cid:1) (cid:15) (cid:15) C − X / / C The horizontal arrows are the obvious inclusions. We explain the vertical arrows,starting with lk ↓ ( X ) ∗ lk ↓ ( X ) → C − X An object either comes from lk ↓ ( X ) and thus is a pair ( Y, Y → X ) with Y anobject in C − X or comes from lk ↓ ( X ) and thus is a pair ( Y, X → Y ) with Y anobject in C − X . In both cases, the object will be sent to Y . Similarly, on the level ofarrows, it is also the canonical projection from lk ↓ ( X ) or lk ↓ ( X ) to C − X . However,for each object ( Y, Y → X ) in lk ↓ ( X ) and each object ( Y ′ , X → Y ′ ) in lk ↓ ( X ),there is another unique arrow in the join. Send this arrow to the composed arrow Y → X → Y ′ . Next, we will define the arrowCoone (cid:0) lk ↓ ( X ) ∗ lk ↓ ( X ) (cid:1) → C Of course, in order to make the diagram commutative, this functor restricted tothe base lk ↓ ( X ) of the coone is the one already defined above. So we have to definethe images of the extra object tip and the extra arrows. Send tip to X . Let ( Y, Y → X ) be an object of lk ↓ ( X ). The arrow from this object to tip is sent tothe arrow Y → X . Similarly, let ( Y, X → Y ) be an object of lk ↓ ( X ). Then thearrow from tip to this object is sent to the arrow X → Y .Our goal is to show that the diagram D becomes a pushout on the level ofclassifying spaces. Unfortunately, this is not always the case. Consider for examplethe groupoid • ⇄ • with two objects and two non-identity arrows which are inverseto each other. In all these cases, however, the situation is even better: Lemma 2.17.
Assume that there is an object A = X and arrows α : X → A and β : A → X . Then the inclusion C − X → C is a homotopy equivalence.Proof. We show that C − X ↓ X is filtered and thus contractible. The lemma thenfollows from Theorem 2.2 and Remark 2.3. Let ( Y, γ ) be an object in C − X ↓ X ,i.e. γ : Y → X is an arrow in C with Y an object in C − X . Set ǫ := γα . Because ofthe assumption Hom C ( X, X ) = { id X } , the arrow αβ : X → X must be the identity.Then we calculate γ = γ ( αβ ) = ( γα ) β = ǫβ This shows that ǫ represents an arrow ( Y, γ ) → ( A, β ) in C − X ↓ X . In particular,for every two objects in the comma category, there are arrows to the object ( A, β ).This shows the first property of a filtered category.For the second property, we have to show that any two parallel arrows are co-equalized by another arrow. So let (
Z, ν ) and (
Y, γ ) be two objects and ǫ, ǫ ′ : ( Z, ν ) → ( Y, γ ) be two arrows, i.e. ǫ, ǫ ′ : Z → Y are arrows in C − X and we have ǫγ = ν = ǫ ′ γ .Set µ = γα which is an arrow ( Y, γ ) → ( A, β ) as already pointed out. Then wecalculate ǫµ = ǫγα = να = ǫ ′ γα = ǫ ′ µ and we are done. (cid:3) In all other cases, diagram D is indeed a pushout on the level of classifyingspaces: Lemma 2.18.
Assume that for any object A = X either there are only arrows from X to A or there are only arrows from A to X , but never both. Then the diagram B ( D ) B (cid:0) lk ↓ X (cid:1) (cid:15) (cid:15) / / Cone (cid:0) B (cid:0) lk ↓ X (cid:1)(cid:1) (cid:15) (cid:15) B (cid:0) C − X (cid:1) / / B ( C ) is a pushout of spaces.Proof. We claim that the nerve functor applied to the diagram D N (cid:0) lk ↓ X (cid:1) (cid:15) (cid:15) / / N (cid:0) Coone (cid:0) lk ↓ X ∗ lk ↓ X (cid:1)(cid:1) (cid:15) (cid:15) N (cid:0) C − X (cid:1) / / N ( C )yields a pushout in SSET . Since the geometric realization functor | ? | : SSET → TOP is left adjoint to the singular simplex functor, it preserves all colimits and inparticular all pushouts. Therefore, applying the geometric realization functor | ? | tothe diagram N ( D ), we obtain a pushout in TOP , as claimed in the lemma.A simplex in N ( C ) is just a string of composable arrows A → . . . → A k . Onecan easily deduce from the assumption that whenever there are two occurences of X in such a string of composable arrows, then there cannot be objects different PERAD GROUPS AND THEIR FINITENESS PROPERTIES 17 from X in between. In other words, if X occurs at all, then all the X in the stringare contained in a maximal substring of the form X → X → . . . → X where all thearrows are (necessarily) id X .Assume now that we have a commutative diagram as follows: N (cid:0) lk ↓ X (cid:1) (cid:15) (cid:15) / / N (cid:0) Coone (cid:0) lk ↓ X ∗ lk ↓ X (cid:1)(cid:1) (cid:15) (cid:15) f ❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇❇ N (cid:0) C − X (cid:1) / / g , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ N ( C ) h ( ( ◗◗◗◗◗◗◗◗ Z We will show that h is uniquely determined by f and g . Assume σ is a simplexin N ( C ) given by a string of composable arrows A → . . . → A k . If all the A i arecontained in the full subcategory C − X then σ is a simplex in the simplicial subset N (cid:0) C − X (cid:1) and then necessarily h ( σ ) = g ( σ ). On the other hand, assume that notall the A i are objects of C − X , i.e. at least one A i = X . As pointed out above, σ must be of the form B → . . . → B r → X → . . . → X → C → . . . → C s where B i = X for all i = 0 , . . . , r and C j = X for all j = 0 , . . . , s . All the B i are in the image of lk ↓ ( X ) because, after composing, we get an arrow B i → X .Analogously, all the C j are in the image of lk ↓ ( X ). Such a simplex σ always liftsto a unique simplex ¯ σ along the map N (cid:0) Coone (cid:0) lk ↓ X ∗ lk ↓ X (cid:1)(cid:1) → N ( C )Thus we have h ( σ ) = f (¯ σ ). This proves uniqueness of h . Showing that h actuallydefines a map of simplicial sets is left to the reader. (cid:3) We combine Lemmas 2.15, 2.17 and 2.18 to get:
Proposition 2.19.
If the descending link lk ↓ ( X ) is ( n − -connected, then thepair ( C , C − X ) is n -connected. More generally, let X be the full subcategory of the category X spanned by acollection of objects in X . Assume that Hom X ( X, X ) = { id X } for all objects X in X \ X . Then X can be built up from X by successively adding objects. If allthe descending links appearing this way are highly connected, then also the pair( X , X ) will be highly connected and, using the long exact homotopy sequence, weobtain the following: Theorem 2.20.
Let x ∈ X be an object. Assume that each descending link is n -connected. Then, we have π k ( X , x ) ∼ = π k ( X , x ) for k = 0 , . . . , n . We say that two objects x and x in X \ X are independent if there are noarrows x → x or x → x in X . This guarantees independence of lk ↓ ( x ) and lk ↓ ( x ) from the adding order of x and x .Again, we can encode the adding order with the help of a Morse function f whichassigns to each object in X \ X an element in a totally ordered set, e.g. N . Werequire that objects with the same f -value are pairwise independent and we addobjects in order of increasing f -values. Operad groups
In this section, we want to introduce our main objects of study, the operadgroups. We first define the types of operads we will be working with. We will thendefine operad groups to be the fundamental groups of the category of operatorsnaturally associated to operads. In the last subsection, we will discuss examplesof operads and their corresponding operad groups. We will recover some alreadywell-known Thompson-like groups this way.3.1.
Basic definitions.Definition 3.1.
An operad O consists of a set of colors C and sets of operations O ( a , . . . , a n ; b ) for each finite ordered sequence a , . . . , a n , b of colors in C (the a i are the input colors and b is the output color) with n ≥ O ( c , . . . , c k ; a ) × . . . × O ( c n , . . . , c nk n ; a n ) × O ( a , . . . , a n ; b ) (cid:15) (cid:15) O ( c , . . . , c k , c , . . . , c nk n ; b )denoted by ( φ , . . . , φ n ) ∗ θ . Composition is associative (Figure 2): (cid:0) ( ψ , . . . , ψ k ) ∗ φ , . . . , ( ψ n , . . . , ψ nk n ) ∗ φ n (cid:1) ∗ θ k ( ψ , . . . , ψ k , ψ , . . . , ψ nk n ) ∗ (cid:0) ( φ , . . . , φ n ) ∗ θ (cid:1) For each color a there are distinguished unit elements 1 a ∈ O ( a ; a ) such that(1 a , . . . , a n ) ∗ θ = θ = θ ∗ b for each operation θ . Sometimes we call such an operad planar in order to distin-guish it from the symmetric or braided versions below.A symmetric/braided operad comes with additional maps (Figure 3) x · : O ( a , . . . , a n ; b ) → O ( a ⊲x , . . . , a n⊲x ; b )for each x in the symmetric group S n or in the braid group B n respectively. Here, i⊲x for x ∈ S n means plugging the element i into the permutation x which isconsidered as a bijection of the set { , . . . , n } . There is a canonical projection B n → S n , so this makes sense also in the braided case. These maps are assumedto be actions: x · ( y · θ ) = ( xy ) · θ · θ = θ They also have to be equivariant with respect to composition (Figure 4):( φ ⊲x , . . . , φ n⊲x ) ∗ ( x · θ ) = ¯ x · (cid:0) ( φ , . . . , φ n ) ∗ θ (cid:1) ( y · φ , . . . , y n · φ n ) ∗ θ = ( y , . . . , y n ) · (cid:0) ( φ , . . . , φ n ) ∗ θ (cid:1) Here, ¯ x is obtained from x by replacing the i ’th strand of x by n i strands and n i isthe number of inputs of φ i⊲x . Furthermore, ( y , . . . , y n ) is the juxtaposition of thepermutations resp. braidings y i . Remark . There is an equivalent way of writing the composition, namely withso-called partial compositions. The i ’th partial compositions ∗ i : O ( c , . . . , c k ; a i ) × O ( a , . . . , a n ; b ) → O ( a , . . . , a i − , c , . . . , c k , a i +1 , . . . , a n ; b )are defined as φ ∗ i θ := (cid:0) a , . . . , a i − , φ, a i +1 , . . . , a n (cid:1) ∗ θ PERAD GROUPS AND THEIR FINITENESS PROPERTIES 19 θ ba a a c c φ c φ c c c φ θ ba a a Figure 1.
Visualization of an operation and composition of operations. ψ ψ ψ φ φ θψ ψ ψ φ φ θ Figure 2.
Associativity. θ ba a a a a a a a Figure 3.
Action of the braid groups on the operations. θφ φ φ φ φ φ θ φ φ φ θφ φ φ θ Figure 4.
First and second equivariance property. φ φ φ Figure 5.
Arrow in S ( O ).Conversely, one could define operads via partial compositions and obtain the usualcomposition from successive partial compositions.The planar operads, symmetric operads and braided operads can be organizedinto categories OP , SYM . OP and BRA . OP respectively. Denote by MON , SYM . MON and
BRA . MON the categories of monoidal categories, symmetric monoidal categories andbraided monoidal categories respectively. There are functorsEnd :
MON −→ OP End :
SYM . MON −→ SYM . OP End :
BRA . MON −→ BRA . OP assigning to each (symmetric/braided) monoidal category C an operad End( C ),called the endomorphism operad. The colors of End( C ) are the objects of C andthe sets of operations are given byEnd( C )( a , . . . , a n ; b ) = Hom C ( a ⊗ . . . ⊗ a n , b )Composition in End( C ) is induced by the composition in C in the obvious way. Theunit element in End( C )( a ; a ) is the identity id a : a → a in C . In the symmetric orbraided case, C comes with additional natural isomorphisms γ X,Y : X ⊗ Y → Y ⊗ X .These can be used to define the action of the symmetric resp. braid groups on thesets of operations. In the theory of operads, these endomorphism operads play animportant role since morphisms of operads O →
End( C )are representations of or algebras over the operad O .The functors End have left adjoints S : OP −→ MON S : SYM . OP −→ SYM . MON S : BRA . OP −→ BRA . MON
The (symmetric/braided) monoidal category S ( O ) is called the category of opera-tors. We will define these categories explicitly. We start with the planar case andthen use it to define the braided case. The symmetric case is similar to the braidedcase.So let O be a planar operad with a set of colors C . The objects of S ( O ) are freewords in the colors, i.e. finite sequences of colors in C . An arrow in S ( O ) is a finitesequence of operations in O : If X , . . . , X n are operations in O , the (ordered) inputcolors of X i are ( c i , . . . , c k i i ) and the output color of X i is d i , then the X i give anarrow ( X , . . . , X n ) : ( c , . . . , c k , c , . . . , c k n n ) → ( d , . . . , d n )in S ( O ). Composition is induced by the composition in the operad O and theidentities are given by the identity operations in O . The tensor product is given byjuxtaposition. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 21 φ φ φ φ φ φ φ φ φ Figure 6.
Equivalence in S ( O ).Now let O be a braided operad with set of colors C . By forgetting the actionof the braid groups, we get a planar operad O pl . The braided monoidal category S ( O ) is a certain product Braid ( C ) ⊠ S ( O pl ). The objects of S ( O ) are once morefinite sequences of colors in C . Arrows in S ( O ) are equivalence classes of pairs( β, X ) ∈ Braid ( C ) × S ( O pl ) consisting of a C -colored braid β and a sequence X = ( X , . . . , X n ) of operations of O where the codomain of β equals the domainof X (Figure 5). The equivalence relation on such pairs is the following: Let ( β, X )be such a pair with X = ( X , . . . , X n ). For each i = 1 , . . . , n let σ i be a C -coloredbraid such that σ i · X i is defined. Let σ := σ ⊗ . . . ⊗ σ n and define σ · ( β, X ) := (cid:0) β ∗ σ − , ( σ · X , . . . , σ n · X n ) (cid:1) We require ( β, X ) and ( β ′ , X ′ ) to be equivalent if there exists a σ as above suchthat ( β ′ , X ′ ) = σ · ( β, X ). In other words, it is the smallest equivalence relationrespecting juxtaposition and which is generated by the relation( β ∗ σ, X ) ∼ ( β, σ · X )with X a single operation. This is visualized in Figure 6.Composition in S ( O ) is defined on representatives ( β, X ) and ( δ, Y ). Looselyspeaking, we push the sequence X of operations through the colored braid δ justas in the definition of equivariance for operads, obtain another colored braid X y δ which is obtained from δ by multiplying the strands according to X and anothersequence of operations X x δ which is obtained from X by permuting the operationsaccording to δ , and finally compose the left and right side in Braid ( C ) and S ( O pl ) φ φ φ ψ ψ φ φ φ ψ ψ φ φ φ ψ ψ Figure 7.
Composition in S ( O ).respectively: ( β, X ) ∗ ( δ, Y ) := (cid:0) β ∗ X y δ, X x δ ∗ Y (cid:1) See Figure 7 for a visualization of this procedure. That this definition is independentof the chosen representatives follows from the equivariance properties of operads.Last but not least, the tensor product is defined on representatives ( β, X ) and( δ, Y ) via juxtaposition, i.e. ( β, X ) ⊗ ( δ, Y ) := ( β ⊗ δ, X ⊗ Y ). The identity arrowsare those represented by a pair of identities. Definition 3.3.
The degree of an operation is its number of inputs. The degree ofan object in S ( O ) is the length of the corresponding color word. The degree of anarrow in S ( O ) is the degree of its domain. A higher degree operation resp. objectresp. arrow is one with degree at least 2. Definition 3.4.
Let O be a planar, symmetric or braided operad and let X be anobject in S ( O ). Then the group π ( O , X ) := π (cid:0) S ( O ) , X (cid:1) is called the operad group associated to O based at X .3.2. Normal forms.
In case O is a planar operad, arrows in S ( O ) are just tensorproducts of operations. In the symmetric and braided case, however, arrows areequivalence classes of pairs ( β, X ). In this subsection, we want to give a normalform of such arrows, i.e. canonical representatives ( β, X ). We will treat the braidedcase, the symmetric case is similar and simpler. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 23
Consider a colored braid β with n strands. The i ’th strand is the strand startingfrom the node with index i ∈ { , . . . , n } . Let S be a subset of the index set { , . . . , n } . Deleting all strands in β other than those with an index in S yieldsanother colored braid β | S . We say that β is unbraided on S if β | S is trivial.Let ( n , . . . , n k ) be a sequence of natural numbers with 1 = n < n < . . . Let [ n , . . . , n k ] be a partition of n and β a colored braid with n strands. Then there is a unique decomposition β = β p ∗ β u into colored braids β p and β u such that β p = β p ⊗ . . . ⊗ β k − p is a tensor product of colored braids β ip with | S i | strands and β u is unbraided with respect to [ n , . . . , n k ] .Proof. Define β ip := β | S i and β u := (cid:16) β | − S ⊗ . . . ⊗ β | − S k − (cid:17) ∗ β Then we have β = β p ∗ β u and β u is unbraided with respect to [ n , . . . , n k ]. Theuniqueness statement is left to the reader. (cid:3) Now let [ β, X ] be an arrow in S ( O ) with X = ( X , . . . , X k ). Assume deg( X i ) = d i and d + . . . + d k = n . Define n i = 1 + P i − j =1 d j for i = 1 , . . . , k + 1 and observethe partition [ n , . . . , n k +1 ]. Decompose the colored braid β − as in the previouslemma to obtain β = τ ∗ ρ where τ − is unbraided with respect to [ n , . . . , n k +1 ]and ρ = ρ ⊗ . . . ⊗ ρ k is a tensor product of colored braids ρ i with d i strands. Define Y i = ρ i · X i . Then from the definition of arrows in S ( O ) it follows that[ β, X ] = [ τ, Y ]with Y = ( Y , . . . , Y k ). So each arrow has a representative ( τ, Y ) such that τ − is unbraided in the ranges defined by the domains of the operations in the secondcomponent. It is easy to see that there is at most one such pair.Similarly, in the symmetric case, for each arrow in S ( O ), there is a uniquerepresentative ( τ, Y ) such that the colored permutation τ − is unpermuted on thedomains of the operations in the second component. Definition 3.6. The unique representative ( τ, Y ) of an arrow in S ( O ) with τ − unpermuted resp. unbraided on the domains of the operations in Y is called thenormal form of that arrow.3.3. Calculus of fractions and cancellation properties. In the following, wewrite θ ≈ ψ if two operations θ, ψ in an operad are equivalent modulo the action ofthe symmetric resp. braid groups, i.e. there exists a permutation resp. braid γ suchthat θ = γ · ψ . Of course, in the planar case, this just means equality of operations. Definition 3.7. Let O be a (symmetric/braided) operad. We say that O satisfiesthe calculus of fractions if the following two conditions are satisfied: • ( Square filling ) For every pair of operations θ and θ with the same outputcolor, there are sequences of operations Ψ = ( ψ , . . . , ψ k ) and Ψ =( ψ , . . . , ψ k ) such that Ψ i ∗ θ i is defined for i = 1 , ∗ θ ≈ Ψ ∗ θ . • ( Equalization ) Assume we have an operation θ and sequences of operationsΨ = ( ψ , . . . , ψ k ) and Ψ = ( ψ , . . . , ψ k ) such that Ψ ∗ θ ≈ Ψ ∗ θ , i.e. thereis a γ with Ψ ∗ θ = γ · (Ψ ∗ θ ). Then γ is already of the form γ = γ ⊗ . . . ⊗ γ k such that γ j · ψ j is defined for each j = 1 , . . . , k and there is a sequence ofoperations Ξ j for each j = 1 , . . . , k such that Ξ j ∗ ψ j = Ξ j ∗ ( γ j · ψ j ). Definition 3.8. Let O be a (symmetric/braided) operad. We define right can-cellativity and left cancellativity for O as follows: • ( Right cancellativity) Assume we have an operation θ and sequences ofoperations Ψ = ( ψ , . . . , ψ k ) and Ψ = ( ψ , . . . , ψ k ) such that Ψ ∗ θ ≈ Ψ ∗ θ , i.e. there is a γ with Ψ ∗ θ = γ · (Ψ ∗ θ ). Then γ is already of theform γ = γ ⊗ . . . ⊗ γ k such that γ j · ψ j is defined and equal to ψ j for each j = 1 , . . . , k . • ( Left cancellativity) Assume we have operations θ and θ and a sequenceof operations Ψ such that Ψ ∗ θ = Ψ ∗ θ . Then θ = θ .We say that O is cancellative if it is both left and right cancellative.These two definitions are designed such that the following two propositions hold.The proofs are straightforward and left to the reader (see also [42]). Proposition 3.9. O satisfies the calculus of fractions if and only if S ( O ) does. Proposition 3.10. O satisfies the left resp. right cancellation property if and onlyif S ( O ) does. Operads with transformations. Observe that the colors of an operad O together with the degree 1 operations form a category I ( O ). In general, this cate-gory could be any category. Thus, to prove certain theorems, it is often necessaryto impose restrictions on the degree 1 operations. Definition 3.11. A planar resp. symmetric resp. braided operad O is called a planar resp. symmetric resp. braided operad with transformations if the category I ( O ) is a groupoid. In other words, all the degree 1 operations are invertible.For such an operad, a transformation is an arrow in S ( O ) of the form [ σ, X ] where X = ( X , . . . , X n ) is a sequence of operations of degree 1. The transformations forma groupoid which we call T ( O ).We say that two operations θ and θ are transformation equivalent if there is atransformation α such that θ = α ∗ θ . We denote by T C ( O ) the set of equivalenceclasses of operations modulo transformation. Note that two transformation equiv-alent operations have the same degree. Thus, we also have a notion of degree forelements in T C ( O ). We define a partial order on the set T C ( O ) as follows: WriteΘ ≤ Θ if there is an operation θ with [ θ ] = Θ and operations ψ , . . . , ψ n suchthat ( ψ , . . . , ψ n ) ∗ θ ∈ Θ . Then, for every θ with [ θ ] = Θ there are operations ψ , . . . , ψ n such that ( ψ , . . . , ψ n ) ∗ θ ∈ Θ . It is not hard to prove that this relationis indeed a partial order. Note that the degree function on T C ( O ) strictly respectsthis order relation which meansΘ < Θ = ⇒ deg(Θ ) < deg(Θ )The following observation, which easily follows from the definitions, reinterpretesthe square filling property of Definition 3.7 in terms of the poset T C ( O ) of trans-formation classes: Observation . Let O be a (symmetric/braided) operad with transformations.Then O satisfies the square filling property if and only if for each pair Θ , Θ oftransformation classes with the same codomain color there is another transforma-tion class Θ with Θ ≤ Θ ≥ Θ . PERAD GROUPS AND THEIR FINITENESS PROPERTIES 25 Spines in graded posets. We call a poset P graded if there is degree functiondeg : P → N such that deg( x ) < deg( y ) whenever x < y . For example, T C ( O )above is graded. Definition 3.13. Let P be a graded poset and M ⊂ P be the subset of minimalelements in P . The spine S of P is the smallest subset S ⊂ P such that M ⊂ S and which satisfies the following property: Whenever v ∈ P \ S , then there is agreatest element g ∈ S such that g < v .We want to prove that the spine of a graded poset always exists. Construction 3.14. We define S i ⊂ P for i ∈ { , , , . . . } inductively. Set S = M . Assume that S i has been constructed. For each pair x, y ∈ S i with x = y ,define M i +1 ( x, y ) ⊂ P to be the set consisting of all the minimal elements z withthe property x ≤ z ≥ y . Now, let S i +1 be the union of all the M i +1 ( x, y ). Finally,define S = S ∞ i =0 S i .In the following, we want to show that this S satisfies the defining properties ofthe spine of P . Observation . Let A ⊂ P . Assume that v ∈ P satisfies a < v for all a ∈ A . Weclaim that there is a minimal element p in the set { z ∈ P | ∀ a ∈ A a ≤ z } which alsosatisfies p ≤ v . If v is already minimal, then we can set p = v . If it is not minimal,there must be another element v ′ ∈ P with a ≤ v ′ < v for all a ∈ A . Then v ′ hasstrictly smaller degree than v . If we repeat this argument with v ′ , we have to endup with a minimal element p at some time, because the degree function is boundedbelow. This p surely satisfies p ≤ v .Let v ∈ P \ S . We want to find the greatest element in the set V := { z ∈ S | z < v } For each i , set S ↓ i = S i ∩ V . We claim: There exists exactly one i such that | S ↓ j | > j < i , | S ↓ i | = 1 and S ↓ j = ∅ for j > i and the unique element in S ↓ i is the greatest element in V .Observation 3.15 applied to A = ∅ reveals that S ↓ = ∅ . Note that either all butfinitely many of the S i are empty or the sequence of numbers d i := min { deg( z ) | z ∈ S i } tends to infinity. But the degree of all the elements in all the S ↓ i is bounded bydeg( v ). It follows that in any case there must be an i such that S ↓ j = ∅ for all j > i . Choose the i which is minimal with respect to this property, i.e. S ↓ i = ∅ .Assume | S ↓ i | > x = y be two elements in this set. Write A = { x, y } and recall that x, y < v . Thus, by Observation 3.15, we know that there must bea p ∈ M i +1 ( x, y ) with p ≤ v . Since v S , we have indeed p < v . Consequently, p ∈ S ↓ i +1 , a contradiction. So we have indeed | S ↓ i | = 1. Next, observe that forany j , if S j = ∅ , then S j − consists of at least two elements. This follows directlyfrom the definitions. Consequently, the same holds for the S ↓ j . From this, it easilyfollows | S ↓ j | > j < i .We now use this to prove that the unique element g ∈ S ↓ i is the greatest elementin V , i.e. x ≤ g whenever x ∈ S with x < v . Let x be such an element. If x = g ,then there must be some j < i such that x ∈ S ↓ j . There is another element x ′ in this S ↓ j . Observation 3.15 applied to A = { x, x ′ } shows that there is p ∈ S ↓ j +1 with x ≤ p . If j + 1 = i , that p must be g and we are done. Else, we repeat thisprocess with p in place of x until we reach level i . This completes the proof that S satisfies the last property in Definition 3.13. Remains to prove that S is the smallest subset containing M and satisfyingthis property. So let S ′ ⊂ P be another subset containing M and satisfying thisproperty. We have to show S ⊂ S ′ . We will prove S i ⊂ S ′ by induction over i . Theinduction start is trivial because S = M . For the induction step, assume S i ⊂ S ′ .Let v ∈ S i +1 . Assume that v S ′ . Then there is a greatest element p ∈ S ′ with p < v . Furthermore, there must be x, y ∈ S i with x = y and v ∈ M i +1 ( x, y ). Thismeans that v is minimal with respect to x ≤ v ≥ y . Since x, y ∈ S ′ but v S ′ wehave indeed x < v > y . Since p is the greatest element in S ′ with p < v , we obtain x ≤ p ≥ y . This contradicts the minimality of v . So we must have v ∈ S ′ and thus S i +1 ⊂ S ′ .3.4.2. Elementary and very elementary operations. Denote by T C ∗ ( O ) the full sub-poset of T C ( O ) spanned by the higher degree classes (i.e. the elements of degree atleast 2). Definition 3.16. Let O be a (symmetric/braided) operad with transformations.The minimal elements in T C ∗ ( O ) are called very elementary transformation classes.Denote the set of very elementary classes by V E .Let Θ , Θ , . . . , Θ k ∈ T C ( O ) be (not necessarily distinct) transformation classes.We say that Θ is decomposable into the classes Θ i if we find operations θ i ∈ Θ i for i = 1 , . . . , k which can be partially composed (see Remark 3.2) in a certain way toan operation in Θ. It can be shown that any class in T C ∗ ( O ) decomposes into veryelementary classes. Definition 3.17. Let O be a (symmetric/braided) operad with transformations.The elements in the spine of T C ∗ ( O ) are called elementary transformation classes.Denote the set of elementary classes by E .An operation in O is called (very) elementary if it is contained in a (very) ele-mentary transformation class. We will call the elementary but not very elementaryclasses resp. operations strictly elementary. Definition 3.18. O is finitely generated if there are only finitely many very ele-mentary transformation classes. It is of finite type if there are only finitely manyelementary transformation classes.The following proposition states that the subsets V E and E are invariant underthe right action of degree 1 operations. Proposition 3.19. Let O be a (symmetric/braided) operad with transformations.Let θ be a higher degree operation and γ be a degree operation. Then the trans-formation class [ θ ] is (very) elementary if and only if the class [ θ ] ∗ γ := [ θ ∗ γ ] is(very) elementary. In particular, the operation θ is (very) elementary if and onlyif θ ∗ γ is (very) elementary.Proof. The main observation is that if Θ , Θ ′ are two transformation classes, thenΘ < Θ ′ holds if and only if Θ ∗ γ < Θ ′ ∗ γ holds. This implies that Θ ∈ V E if andonly if Θ ∗ γ ∈ V E or, in other words, V E ∗ γ = V E . Now write E ′ = E ∗ γ . Wethen have V E ⊂ E ′ . Let Θ ∈ T C ∗ ( O ) \ E ′ . Then Θ ∗ γ − ∈ T C ∗ ( O ) \ E . Thus,by the definition of E as the spine of T C ∗ ( O ), we have that there is a greatestelement Ψ ∈ E with Ψ < Θ ∗ γ − . Then Ψ ∗ γ ∈ E ′ is the greatest element withΨ ∗ γ < Θ. Consequently, E ′ satisfies the defining properties of the spine E . Itfollows E ⊂ E ′ = E ∗ γ . Since this holds for arbitrary γ , we obtain E = E ∗ γ . (cid:3) Examples. In this subsection, we want to present some examples of operadsleading to already well-known operad groups as well as to new groups to which thefiniteness result of Section 4 is applicable. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 27 Free operads. Only very briefly we want to remark that the operad groupsassociated to operads freely generated by operations of degree at least 2 correspondexactly to the so-called diagram groups defined in [25]. When considering free symmetric operads, we get symmetric versions of diagram groups which are called“braided” in [25, Definition 16.2]. The truly braided diagram groups are the onesarising from free braided operads.In particular, the operad group associated to the operad O F freely generated byone color and a single binary operation is isomorphic to Thompson’s group F . Thishas first been observed in [18]. Moreover, if we consider the symmetric resp. braidedoperad O V resp. O BV freely generated by one color and a single binary operation,we obtain Thompson’s group V resp. the braided Thompson group BV .More details on the free case can be found in [42].3.5.2. Suboperads of endomorphism operads. Recall that there is a planar resp. sym-metric resp. braided operad End( C ) naturally associated to each planar resp. sym-metric resp. braided monoidal category C , called the endomorphism operad. Thecolors of End( C ) are the objects in C and the sets of operations are given byEnd( C )( a , . . . , a n ; b ) = Hom C ( a ⊗ . . . ⊗ a n , b )Let G be a subgroupoid of C and S be a set of higher degree operations in E :=End( C ) with outputs and inputs being objects of G . Then we can look at thesuboperad of E generated by this data: It is the smallest suboperad O such that I ( O ) = G and such that the elements in S are operations in O . These suboperadsare in general not free in the sense of 3.5.1 though we have only specified generators.The relations are automatically modelled by the ambient category C .Not always is the map S → T C ( O ) sending an operation to its transformationclass a bijection onto the set of very elementary classes. However, this is true if thefollowing conditions are satisfied:( V ) If θ, θ ′ ∈ S with θ = θ ′ , then [ θ ] and [ θ ′ ] are incomparable, i.e. [ θ ] [ θ ′ ]and [ θ ] [ θ ′ ]. In particular, they are not equal.( V ) The set of transformation classes represented by operations in S is closedunder right multiplication with operations in G , i.e. for each θ ∈ S and γ ∈ G there is θ ′ ∈ S with [ θ ∗ γ ] = [ θ ′ ].We want to be a bit more explicit now and observe suboperads of the endo-morphism operad E of the symmetric monoidal category ( TOP , ⊔ ) where ⊔ is thecoproduct (i.e. the disjoint union) of topological spaces. We call an operation( f , . . . , f k ) in E mono if the images of the maps f i : X i → X are pairwise disjointin X and the f i are injective. We call it epi if the images cover X . It is not hardto prove that if all operations in a suboperad of E are mono, then it satisfies theright cancellation property. Likewise, if all the operations are epi, then it satisfiesthe left cancellation property.We give an explicit example to illustrate the above procedure. Consider the unitsquare and the right angled triangle obtained by halving the unit square:Consider all isometries of the square and the triangle, i.e. the dihedral group D and Z / Z . The disjoint union of these isometry groups forms a groupoid G lyingin TOP . Consider the following subdivisions, called very elementary subdivisions: The set S has three elements, one for each sudivision: The first one maps foursquares to each square in the first subdivision via coordinate-wise linear trans-formations. The second one maps four triangles to each triangle in the secondsubdivision via orientation preserving similarities. The third one maps two trian-gles to each triangle in the third subdivision via orientation preserving similarities.As above, the groupoid G together with the set S generate a suboperad O of thesymmetric operad E = End( TOP , ⊔ ).The transformation classes are in one to one correspondence with subdivisions ofthe square or the triangle which can be obtained by iteratively applying the threesubdivisions above. We have Θ ≤ Θ if and only if Θ can be obtained from Θ by performing further subdivisions. For example, we have ≤ From this it follows easily that the transformation classes represented by the veryelementary subdivisions are not comparable, i.e. ( V ) is satisfied. Furthermore,when applying an isometry of the square or the triangle to one of the operationsin S , we obtain the same operation with a transformation precomposed. Thus,also ( V ) is satisfied. It follows that the very elementary subdivisions correspondexactly to the very elementary classes of O .To find all the elementary transformation classes, we have to follow the construc-tion in 3.14. There is exactly one minimal subdivision of the square which refinesthe two very elementary subdivisions of the square:Thus, this subdivision represents the only elementary class which is not very ele-mentary.All the operations in O are clearly epi, so it satisfies the left cancellation property.Not all of them are mono, but we can change the definitions a little bit and obtain anisomorphic operad where all operations are mono: Instead of the closed square andtriangle, we can consider the open square and triangle and also subdivisions intoopen squares and triangles. Thus, O also satisfies the right cancellation property.Moreover, we claim that it satisfies square filling. To see this, consider the followingchains of subdivisions: · · ·· · · These are cofinal in the sense that every subdivision of the square resp. triangle issmaller than or equal to one of the subdivisions of the first resp. second chain. FromObservation 3.12 it follows that O satisfies square filling. All in all, O satisfies thecancellative calculus of fractions. ⊲ Cube cutting operads. Let N be a finite set of natural numbers greater thanor equal to 2. Denote by h N i the multiplicative submonoid of N generated bythe numbers in N . We say that the numbers in N are independent if, whenevera natural number n can be written as a product n r · · · n r k k of pairwise distinctnumbers n i ∈ N , then the exponents r i are already uniquely determined by n . In PERAD GROUPS AND THEIR FINITENESS PROPERTIES 29 other words, N is a basis for h N i . This is satisfied for example if the numbers in N are pairwise coprime or, even stronger, if they are prime. For later reference, werecord the following two trivial observations:( B ) No number n ∈ N is a product of other numbers in N .( B ) Whenever n , . . . , n k ∈ N are pairwise distinct numbers and n ∈ h N i isdivisible by each n i in h N i , i.e. there is m i ∈ h N i with n = n i m i , then n isalso divisible by the product n · · · n k in h N i .There are non-bases N which satisfy ( B ) but not ( B ), for example N = { , , , } .For this N we have 6 · · d ≥ d -dimensional unit cube and a subgroup if its group of isometries.Define this group to be the groupoid G lying in TOP . Next, we want to specify veryelementary subdivisions of the cube. For each j ∈ { , . . . , d } , let N j ⊂ N be a set ofnatural numbers as in the preceding paragraph. For each such j and n ∈ N j , thereis a very elementary subdivision of the cube given by cutting it, perpendicularlyto the j ’th coordinate axis, into n congruent subbricks. The following are the veryelementary subdivisions in the case d = 2, N = { } and N = { } :There is one operation in S for each such very elementary subdivision: Cubes arecoordinate-wise linearly rescaled to fit into the subbricks of the subdivisions. Thegroupoid G together with the set S generate a suboperad O of E = End( TOP , ⊔ )which we call a symmetric cube cutting operad since we will also define planar cubecutting operads below.The transformation classes are in one to one correspondence with subdivisionsof the cube obtained by iteratively applying n -cuts in direction j as above. Twotransformation classes are comparable if and only if one is a subdivision of theother. From ( B ) it follows that two very elementary subdivisions are not compa-rable. Consequently, ( V ) is satisfied. It is not always true that right multiplicationof elements in G with operations in S yields another operation in S up to transfor-mation. For example, a rotation of a vertically cutted square by an angle of π/ V ) is satisfied or not depends on theinterplay between the isometries in G and the sets N j . For example, it is satisfied if G = 1 or if N = . . . = N d . Let us always assume that G and the N j are compatiblein a way such that ( V ) is satisfied. Then the very elementary subdivisions are inone to one correspondence with the very elementary transformation classes.We want to identify the elementary transformation classes. For each element T = ( T , . . . , T d ) ∈ N × . . . × N d of the product of the power sets such that T = ( ∅ , . . . , ∅ ), there is a transformation class Θ T which is obtained by iterativelyperforming, for each j ∈ { , . . . , d } and each n ∈ T j , an n -cut in direction j onevery subbrick. The result is independent of the order of the cuts. These classesare exactly the elementary classes. To see this, we make the following claim: IfΘ T and Θ T ′ are two such classes, then Θ T ∪ T ′ is the smallest class Θ satisfyingΘ T ≤ Θ ≥ Θ T ′ . Here, the inclusion T ⊂ T ′ and the union T ∪ T ′ is meant to becoordinate-wise. The figure below pictures the elementary operations in the case d = 2, N = { , } and N = { , } . E E E E To see the above claim, we consider the case d = 1 and set N := N . The case d > T ≤ Θ ≥ Θ T ′ . It is not hard to find the greatest regular class Θ r withΘ r ≤ Θ. Since Θ T and Θ T ′ are regular, we have Θ T ≤ Θ r ≥ Θ T ′ . There is aunique n ∈ h N i such that n is the length of the subintervals in the subdivision ofΘ r . Then Θ T ≤ Θ r means that the product of the numbers in T divides n in h N i .In particular, each t ∈ T divides n in h N i . Likewise, each t ′ ∈ T ′ divides n in h N i .It follows from ( B ) that the product of the numbers in T ∪ T ′ divides n in h N i .This implies Θ T ∪ T ′ ≤ Θ r and it follows Θ T ∪ T ′ ≤ Θ, q.e.d.All the operations in O are epi and O is isomorphic to a suboperad of E whereall operations are mono by considering open cubes instead of closed ones. Conse-quently, O satisfies the left and right cancellation property. We also find a cofinalchain of subdivisions: The first subdivision in this chain is obtained by iterativelyapplying, for each j ∈ { , . . . , d } and each n ∈ N j , an n -cut in direction j on everysubbrick. Then the whole chain is obtained by iterating this with every subbrick.For example, in the case d = 2, N = { } and N = { } , we can take the followingchain: · · · Thus, O satisfies the square filling property. All in all, it satisfies the cancellativecalculus of fractions.Note that the symmetric cube cutting operads are symmetric operads with trans-formations. When forgetting the symmetric structure on E , we obtain a planaroperad E pl and we can define suboperads, which are then planar operads withtransformations and which we call planar cube cutting operads, as follows: Con-sider the case d = 1. Set G = 1. Let N ⊂ N be a set of natural numbers as inthe first paragraph. There is one very elementary subdivision of the unit inter-val for each n ∈ N , cutting it into n pieces of equal length. The operations in S linearly map unit intervals to the subintervals of very elementary subdivisions.This time, however, we specify the order of these maps. We require that they areordered by their images via the natural ordering on the unit interval. Denote by O the suboperad of E pl generated by this data. Note that O is a planar operadwith transformations which is degenerate in the sense that there are no degree 1operations besides the identities. Thus, a transformation class is the same as an op-eration. Operations in O are in one to one correspondence with subdivision of theunit interval which are obtained by iteratively applying n -cuts for various n ∈ N .Two operations are related if and only if one is a subdivision of the other. The very PERAD GROUPS AND THEIR FINITENESS PROPERTIES 31 elementary operations are in one to one correspondence with the very elementarysubdivisions and the elementary operations can be described just as in the case ofsymmetric cube cutting operads. Furthermore, O satisfies the cancellative calculusof fractions.We now look at the operad groups associated to these planar resp. symmetriccube cutting operads. Using the fact that arrows in the fundamental groupoid of acategory satisfying the calculus of fractions can be represented by spans, it is easyto identify the following operad groups (where G = 1 in each case): • The Higman-Thompson groups F n,r resp. V n,r arise as the operad groups(based at the object represented by a disjoint union of r unit intervals)associated to the planar resp. symmetric cube cutting operads with d = 1and N = { n } . • The groups of piecewise linear homeomorphisms of the (Cantor) unit inter-val F (cid:0) r, Z [ n ··· n k ] , h n , . . . , n k i (cid:1) resp. G (cid:0) r, Z [ n ··· n k ] , h n , . . . , n k i (cid:1) consid-ered in [39] arise as the operad groups (based at the object represented bya disjoint union of r unit intervals) associated to the planar resp. symmetriccube cutting operads with d = 1 and N = { n , . . . , n k } . • The higher dimensional Thompson groups nV (see [4]) arise as the operadgroups (based at the object represented by the n -dimensional unit cube)associated to the symmetric cube cutting operads with d = n and N j = { } for all j = 1 , . . . , d . ⊲ Local similarity operads. In [26] groups were defined which act in a certainway on compact ultrametric spaces. We recall the definition of a finite similaritystructure: Definition 3.20. Let X be a compact ultrametric space. A finite similarity struc-ture Sim X on X consists of a finite set Sim X ( B , B ) of similarities B → B forevery ordered pair of balls ( B , B ) such that the following axioms are satisfied: • ( Identities ) Each Sim X ( B, B ) contains the identity. • ( Inverses ) If γ ∈ Sim X ( B , B ), then also γ − ∈ Sim X ( B , B ). • ( Compositions ) If γ ∈ Sim X ( B , B ) and γ ∈ Sim X ( B , B ), then also γ γ ∈ Sim X ( B , B ). • ( Restrictions ) If γ ∈ Sim X ( B , B ) and B ⊂ B is a subball, then also γ | B ∈ Sim X ( B , γ ( B )).Here, a similarity γ : X → Y of metric spaces is a homeomorphism such thatthere is a λ > d ( γ ( x ) , γ ( x )) = λd ( x , x ) for all x , x ∈ X . Let Sim X be afinite similarity structure on the compact ultrametric space X . A homeomorphism γ : X → X is said to be locally determined by Sim X if for every x ∈ X there is aball x ∈ B ⊂ X such that γ ( B ) is a ball and γ | B ∈ Sim X ( B, γ ( B )). The set of allsuch homeomorphisms forms a group which we denote by Γ(Sim X ).To a finite similarity structure Sim X , we can associate a symmetric operad withtransformations O , a suboperad of E = End( TOP , ⊔ ), and reobtain the groupsΓ(Sim X ) as operad groups. We do this by appealing to the procedure above. Twoballs B , B in X are called Sim X -equivalent if Sim X ( B , B ) = ∅ . Choose one ballin each Sim X -equivalence class (the isomorphism class of the operad we will definedoes not depend on this choice). Consider the groupoid G lying in TOP which is thedisjoint union of the groups Sim X ( B, B ) with B a chosen ball. The set S containsone operation in E for each chosen ball B : Consider the maximal proper subballs A , . . . , A k of B . For each i = 1 , . . . , k choose a similarity γ i ∈ Sim X ( B i , A i ) where B i is the unique chosen ball equivalent to A i . Now the operation associated to B maps the chosen balls B i to A i using the similarities γ i . The data ( G , S ) generatesa suboperad O of E . Each transformation class in O is uniquely determined by a chosen ball togetherwith a subdivision into subballs. Two such subdivisions are related if and only if onecan be obtained from the other by further subdividing the subballs. Condition ( V )is trivially true since the operations in S have different codomains. A similarity inSim X ( B, B ) is an isometry γ : B → B which permutes the maximal proper subballsand the restriction of γ to a maximal proper subball is again a similarity in Sim X . Itfollows that right multiplication of an element in G with an operation in S gives thesame operation modulo transformation. In particular, ( V ) is satisfied. Thus, thevery elementary classes of O are in one to one correspondence with the chosen ballstogether with their subdivisions into the proper maximal subballs. Since every twovery elementary classes have different colors as codomains, there are no elementaryclasses which are not very elementary.All the operations in O are both mono and epi. Thus, it satisfies both left andright cancellation. It also satisfies square filling and thus the cancellative calculusof fractions since we again find a cofinal sequence of subdivisions for each chosenball B : Define the chain inductively by subdividing each subball by their maximalproper subballs.Using the fact that arrows in the fundamental groupoid of a category satisfyingthe calculus of fractions can be represented by spans, it is not hard to establish anisomorphism π ( O , X ) ∼ = Γ(Sim X ) where we assume that X is the chosen ball ofits Sim X -equivalence class.3.5.3. Ribbon Thompson group. To close this subsection, we briefly want to discussan operad yielding an operad group RV which naturally fits into the sequence ofwell-known groups F, V, BV . First observe the free braided operad with transfor-mations generated by a single color, the group Z as groupoid of degree 1 operationsand a single binary operation. The components of the corresponding groupoid oftransformations are the groups B n ⋉ Z n . Think of elements of these groups asribbons which can braid and twist. A single twist corresponds to a generator in Z .Then we impose the following relation on this operad:=The caret corresponds to the generating binary operation. The operations in thisbraided operad with transformations are in one to one correspondence with binarytrees together with braiding and twisting ribbons attached to the leaves. Thetransformation classes are in one to one correspondence with binary trees. Theonly very elementary class is represented by the binary tree with two leaves (thecaret). There are no strictly elementary classes. It satisfies the cancellative calculusof fractions. Consequently, elements in the associated operad group based at 1 canbe represented by pairs of binary trees where the leaves are connected by braidingand twisting ribbons. Composition is modelled by concatenating two such treepair diagrams, removing all dipoles formed by carets and then applying the aboverelation in order to obtain another tree pair diagram.4. A topological finiteness result Before we state the main theorem of this article, we have to introduce two moredefinitions. Definition 4.1. We say that a group G is of type F + ∞ if G and all of its sub-groups are of type F ∞ . We then say that a groupoid is of type F + ∞ (or F ∞ ) if itsautomorphism groups are of type F + ∞ (or F ∞ ). PERAD GROUPS AND THEIR FINITENESS PROPERTIES 33 For example, all finite groups and Z are of type F + ∞ . Definition 4.2. Let O be a (symmetric/braided) operad with transformations. Anobject X in S ( O ) is called reduced if no non-transformation arrow in S ( O ) has X as its domain. We call O color-tame if the degree of all reduced objects is boundedfrom above.Note that if O is monochromatic and there exists at least one higher degreeoperation, then it is automatically color-tame. Theorem 4.3. Let O be a planar or symmetric or braided operad with transfor-mations. Assume that O has only finitely many colors and is color-tame. Assumefurther that O satisfies the cancellative calculus of fractions, is of finite type and I ( O ) is a groupoid of type F + ∞ . Then for every object X in S ( O ) the operad group π ( O , X ) is of type F ∞ .Question . Can the requirement color-tameness be dropped? Remark . There is also a version of this theorem for free operads: Assume that O is free as a (symmetric/braided) operad with transformations, has only finitelymany colors, is color-tame, finitely generated and that I ( O ) is a groupoid of type F ∞ . Then π ( O , X ) is of type F ∞ .The proof of the free case is parallel to the proof in this article (with smallmodifications and additions, see [42]). Parts of the theorem for the free case arealso proven in [14, 15] (in the language of diagram groups).Concerning the examples in Subsection 3.5, it should be noted that the freeoperads O F , O V and O BV also satisfy the conditions in Theorem 4.3.The main tool to prove this theorem is, as usual, Brown’s criterion [7]. Moreprecisely, we will need the following special version of it: Theorem 4.6. Let Γ be a discrete group and X be a contractible Γ -CW-complexwith isotropy groups of type F ∞ . Assume we have a filtration ( X n ) n ∈ N of X suchthat each X n is a Γ -CW-subcomplex of finite type and such that the connectivity ofthe pairs ( X n , X n − ) tends to infinity as n → ∞ . Then Γ is of type F ∞ . We sketch a geometric proof of this criterion using a blow-up construction ofL¨uck [29, Lemma 4.1]: For each conjugacy class [ H ] of isotropy groups of X , wechoose a free contractible H -CW-complex EH of finite type. Using these, wecan construct a free Γ-CW-complex F ( X ) which is homotopy equivalent to X .The idea is to replace the equivariant cell Γ /H × D n in X by the Γ-CW-complex(Γ × H EH ) × D n . More details can be found in the proofs of [29, Lemma 4.1 andTheorem 3.1]. We can also apply this construction to each Γ-CW-subcomplex X n and obtain free Γ-CW-complexes F ( X i ) homotopy equivalent to X i and of finitetype. For each n ∈ N we find k ∈ N big enough so that X k and hence F ( X k )is n -connected. By equivariantly gluing cells in dimensions n + 2 and higher, weobtain a free contractible Γ-CW-complex with finitely many equivariant cells up todimension n + 1. Consequently, Γ is of type F n +1 . Since n was arbitrary, it followsthat Γ is of type F ∞ (see e.g. [23, Proposition 7.2.2]).The remaining subsections are devoted to the proof of Theorem 4.3.4.1. Three types of arc complexes. Let d ∈ { , , } and C be a set of colors.Let X = ( c , . . . , c n ) be a word in the colors of C . An archetype consists of a uniqueidentifier together with a word in the colors of C of length at least 2. Let A be a setof archetypes. To this data, we will associate a simplicial complex AC d ( C, A ; X ).Consider the points 1 , . . . , n ∈ R and embed them into R d via the first componentembedding R → R d . Color these points with the colors in the word X (i.e. the point i is colored with the color c i ) and call them nodes. Denote the set of nodes by N .A link is the image of an embedding γ : [0 , → R d such that γ (0) and γ (1) arenodes. Note that a link may contain more than two nodes. Two links connectingthe same set of nodes are equivalent if there is an isotopy of R d \ N which takes onelink to the other. An equivalence class of links is called an arc. Note that in thecase d = 1, arcs and links are the same since each arc is represented by a uniquelink. We say that two arcs are disjoint if there are representing links which aredisjoint. In the cases d = 2 , 3, we can choose representing links of a collection ofarcs such that the links are in minimal position: Lemma 4.7. Assume d = 2 or d = 3 . Let a , . . . , a k be arcs with a i = a j for each i = j . Then there are representing links α , . . . , α k such that | α i ∩ α j | is finite andminimal for each i = j .Proof. In the case d = 3, we can always find representing links which only intersectat nodes, if at all. The case d = 2 is a bit more complicated. We use the ideas from[9, Lemma 3.2]: Consider the nodes as punctures in the plane R . Then we canfind a hyperbolic metric on that punctured plane. Now define α i to be the geodesicwithin the class a i . (cid:3) A link connecting a set of nodes M is called admissible if there is an isotopy of R d \ M taking the link into the image of the first component embedding R → R d .In the case d = 1, this is vacuous. In the case d = 2, this implies in particularthat, when travelling the link starting from the lowest node, the nodes are visitedin ascending order. This last property is even equivalent to being admissible in thecase d = 3. An arc is called admissible if one and consequently all of its links areadmissible. Now label an admissible arc with the identifier of an archetype in A .We require that the word formed by the colors of the connected nodes (in ascendingorder) equals the color word of the archetype. Call such a labelled admissible arcan archetypal arc.The vertices of AC d ( C, A ; X ) are the archetypal arcs. Two vertices are joinedby an edge if the corresponding arcs are disjoint. This determines the complex asa flag complex. A k -simplex is therefore a set of k + 1 pairwise disjoint archetypalarcs. We call this an archetypal arc system. It follows from Lemma 4.7 abovethat if { a , . . . , a k } is an archetypal arc system, then we always find representinglinks α i of a i such that the α i are pairwise disjoint. The following are examples of2-simplices in the cases d = 1 , , non -examples of simplices in the case d = 2. In the first diagram,the two arcs are not disjoint and in the second diagram, the arc is not admissible.However, the second diagram would represent an admissible arc in the case d = 3. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 35 Definition 4.8. Let C be a set of colors and A be a set of archetypes. A wordin the colors of C is called reduced if it admits no archetypal arc on it. The setof archetypes A is called tame if the length of all reduced words is bounded fromabove. The length of an archetype is the length of its color word. The set ofarchetypes A is of finite type if the length of all archetypes is bounded from above. Theorem 4.9. Let d ∈ { , , } . Let C be a set of colors and A be a set ofarchetypes. Assume that A is tame and of finite type. Let m r be the smallestnatural number greater than the length of any reduced color word and m a be themaximal length of archetypes in A . Define ν κ ( l ) := (cid:22) l − m r κ (cid:23) − Let X be a word in the colors of C and denote by lX the length of X . Then thecomplex AC d ( C, A ; X ) is ν κ d ( lX ) -connected where κ := 2 m a + m r − κ := 2 m a − κ := 2 m a − d = 2 we have to pass to a slightly larger class ofcomplexes: Instead of R we consider links and arcs in the punctured plane S = R \ { p , . . . , p l } with finitely many punctures p i ∈ R disjoint from the nodes.Here, we define two links connecting the same set of nodes to be equivalent ifthey differ by an isotopy of S \ N and a link connecting a set of nodes M to beadmissible if there is an isotopy of R \ M taking the link into the image of the firstcomponent embedding R → R . Note that in the latter case, we require an isotopyof R \ M and not of S \ M , i.e. we allow the links to be pulled over punctures. Wedenote the corresponding complex of archetypal arc systems again by AC d ( C, A ; X ),suppressing the additional data of punctures since, as we will see, Theorem 4.9 isstill valid for this larger class of complexes.4.1.1. Proof of the connectivity theorem. The proof essentially consists of slightlymodified ideas from [9, Subsection 3.3].We induct over the length lX of X . The induction start is lX ≥ m r . This impliesthat X is not reduced and thus admits an archetypal arc on it. It follows that AC d ( C, A ; X ) is non-empty, i.e. ( − lX ≥ m r + κ d . We look at the cases d = 1 , , d = 2 since it is the hardest one. ⊲ The two-dimensional case. Choose a vertex of AC := AC ( C, A ; X ) repre-sented by an archetypal arc b . Let v < . . . < v t be the nodes connected by b . Let AC be the full subcomplex of AC spanned by the archetypal arcs which do notmeet the nodes v i .We want to estimate the connectivity of the pair ( AC , AC ) using the Morsemethod for simplicial complexes (see e.g. Subsection 2.10). Let a be an archetypalarc. Define s i ( a ) := ( a meets v i for each i = 1 , . . . , t . Now set h ( a ) := (cid:0) s ( a ) , . . . , s t ( a ) (cid:1) Note that the right side is a sequence of t numbers in { , } . Interpret thesesequences as binary numbers and order them accordingly. Then h is a Morsefunction building up AC from AC since archetypal arcs with h -value equal to(0 , . . . , 0) are exactly the archetypal arcs in AC and two archetypal arcs with thesame h -value different from (0 , . . . , 0) are not connected by an edge.We want to inspect the descending links with respect to this Morse function h .Let a be an archetypal arc with Morse height greater than (0 , . . . , τ ∈ { , . . . , t } such that s τ ( a ) = 1. It is not hard to prove that lk ↓ ( a )is the full subcomplex of AC spanned by archetypal arcs disjoint from a and notmeeting any v i with i < τ . Let X ′ be the color word which is obtained from X by removing the colors corresponding to nodes which are contained in a and tothe nodes v i with i < τ . Then we see that lk ↓ ( a ) is isomorphic to AC ( C, A ; X ′ )with an additional puncture corresponding to a and further additional puncturescorresponding to the nodes v i with i < τ . By induction, it follows that lk ↓ ( a ) is ν κ ( lX ′ )-connected. Denote by l a the length of a , i.e. the number of nodes it meets.Then we can estimate lX ′ = lX − l a − ( τ − ≥ lX − l a − t + 1 ≥ lX − m a − t + 1 ≥ lX − m a + 1Thus, lk ↓ ( a ) is ν κ ( lX − m a + 1)-connected. Consequently, by the Morse method,the connectivity of the pair ( AC , AC ) is ν κ ( lX − m a + 1) + 1 = ν κ ( lX )because of κ = 2 m a − ι : AC → AC induces the trivial map in π m for m ≤ ν κ ( lX ). It then follows from the long exacthomotopy sequence of the pair ( AC , AC ) that AC is ν κ ( lX )-connected whichcompletes the proof in the case d = 2.Let ϕ : S m → AC be a map with m ≤ ν κ ( lX ). We have to show that ψ := ϕ ∗ ι : S m → AC is homotopic to a constant map. Think of S m as the boundaryof an ( m + 1)-simplex. By simplicial approximation [37, Theorem 3.4.8] we cansubdivide S m and homotope ϕ to a simplicial map. So we will assume in thefollowing that ϕ is simplicial. Next, we want to apply [9, Lemma 3.9] in orderto subdivide S m further and homotope ψ to a simplexwise injective map. Thismeans that whenever vertices v = w in S m are joined by an edge, then ψ ( v ) = ψ ( w ). To apply the lemma, we have to show that the link of every k -simplex σ in AC is ( m − k − a , . . . , a k be pairwise disjoint archetypalarcs representing a k -simplex σ . The link of this simplex is the full subcomplexspanned by the archetypal arcs which are disjoint from every a i . Deleting everycolor corresponding to nodes which are contained in one of the a i from X , weobtain a color word X ′ and it is easy to see that the link of σ is isomorphic to AC ( C, A ; X ′ ) with one additional puncture for each a i . By induction, we obtainthat lk ( σ ) is ν κ ( lX ′ )-connected. We have the estimate lX ′ ≥ lX − ( k + 1) m a andthus ν κ ( lX ′ ) ≥ ν κ (cid:0) lX − ( k + 1) m a (cid:1) = (cid:22) lX − ( k + 1) m a − m r m a − (cid:23) − PERAD GROUPS AND THEIR FINITENESS PROPERTIES 37 = (cid:22) lX − m r m a − − ( k + 1) m a m a − (cid:23) − ≥ (cid:22) lX − m r m a − − (2 k + 2)(2 m a − m a − (cid:23) − ν κ ( lX ) − (2 k + 2) ≥ m − k − ψ is simplexwise injective.We now want to show that ψ can be homotoped so that the image is containedin the star of b . Since the star of a vertex is always contractible, this will finish theproof. We will homotope ψ by moving single vertices of S m step by step, eventuallylanding in the star of b . Consider the vertices a , . . . , a l of ψ ( S m ) which do net yetlie in the star of b , i.e. which are not disjoint to b . Choose representing links α i of a i and β of b such that the system of links ( β, α , . . . , α l ) is in minimal positionas in Lemma 4.7. Note the little subtlety that archetypal arcs may have the sameunderlying arc but are different because they have different labels. In this case,homotope the corresponding links a little bit so that they intersect only at nodes.Note also that each α i intersects β , but not at nodes since each a i comes from AC .Last but not least, we can assume that whenever p is an intersection point of β with one of the α i , then there is at most one α i meeting the point p .Now look at the intersection point p of one of the α i with β which is closestto v along β . Write α for the link which intersects β at this point and a for thecorresponding arc. α βw w ′ v j v j +1 Choose a vertex x in S m which maps to a via ψ . Define another link α ′ as follows:Let j be such that the intersection point p lies on the segment of β connecting v j with v j +1 . Denote by w < w ′ the nodes such that p lies on the segment of α connecting w with w ′ . Now push this segment of α along β over the node v j suchthat α and α ′ bound a disk whose interior does not contain any puncture or nodeother than v j . α ′ β Note that α ′ is still admissible. Denote by a ′ the archetypal arc with link α ′ andthe same label as a . Our goal is now to homotope ψ to a simplicial map ψ ′ suchthat ψ ′ ( x ) = a ′ and ψ ′ ( y ) = ψ ( y ) for all other vertices y . Iterating this procedureoften enough, we arrive at a map ψ ∗ homotopic to ψ such that ψ ∗ ( y ) ∈ st ( b ) foreach vertex y . For example, the next step would be to move x to the vertex α ′′ : α ′′ β By simplexwise injectivity, no vertex of lk ( x ) is mapped to a . Furthermore, avertex of AC in the image of ψ disjoint to a must also be disjoint to a ′ becausewe have chosen α such that no other α i intersects β between p and v . From theseobservations, it follows that ψ (cid:0) lk ( x ) (cid:1) ⊂ lk ( a ) ∩ lk ( a ′ )This inclusion enables us to define a simplicial map ψ ′ : S m → AC with ψ ′ ( x ) = a ′ and ψ ′ ( y ) = ψ ( y ) for all other vertices y . Let X ′ be the color word obtained from X by removing all colors corresponding to nodes which are contained in a or tothe node v j . Then lk ( a ) ∩ lk ( a ′ ) is isomorphic to AC ( C, A ; X ′ ) with an additionalpuncture corresponding to the disk bounded by α ∪ α ′ . Thus, by induction, it is ν κ ( lX ′ )-connected. We have the estimate lX ′ ≥ lX − m a − ν κ ( lX ′ ) ≥ ν κ (cid:0) lX − m a − (cid:1) = (cid:22) lX − m a − − m r m a − (cid:23) − (cid:22) lX − m r m a − − m a + 12 m a − (cid:23) − ≥ (cid:22) lX − m r m a − − m a − m a − (cid:23) − ν κ ( lX ) − ≥ m − lk ( x ) is an ( m − lk ( a ) ∩ lk ( a ′ ) impliesthat the map ψ | lk ( x ) : lk ( x ) → lk ( a ) ∩ lk ( a ′ ) can be extended to the star st ( x ) of x which is an m -disk. So we obtain a map ϑ : st ( x ) → lk ( a ) ∩ lk ( a ′ ) coinciding with ψ on the boundary lk ( x ). We can now homotope ψ | st ( x ) rel lk ( x ) to ϑ within st ( a )and further to ψ ′ within st ( a ′ ). This finishes the proof of the theorem in the case d = 2. ⊲ The three-dimensional case. Choose an archetypal arc b connecting the nodes v < . . . < v t and let AC be the full subcomplex of AC := AC ( C, A ; X ) spannedby the archetypal arcs which do not meet the nodes v i .With a very similar Morse argument as in the case d = 2 above, we can showthat the pair ( AC , AC ) is ν κ ( lX )-connected.Again, the second step consists of showing that the inclusion ι : AC → AC induces the trivial map in π m for m ≤ ν κ ( lX ). This is much easier in the case d = 3: Let ϕ : S m → AC be a map and assume without loss of generality that it issimplicial. But then the map ψ := ϕ ∗ ι : S m → AC already lies in the star st ( b )of b since an archetypal arc not meeting any of the nodes v i is already disjoint to b . Consequently, ψ can be homotoped to a constant map and this concludes theproof in the case d = 3. ⊲ The one-dimensional case. Choose an archetypal arc b connecting the nodes v < . . . < v t such that the color word formed by the first r nodes w < v isreduced. Let AC be the full subcomplex of AC := AC ( C, A ; X ) spanned by thearchetypal arcs which do not meet the nodes v i . This condition is equivalent to not PERAD GROUPS AND THEIR FINITENESS PROPERTIES 39 meeting any nodes w ≤ v t . These are simply the first s nodes w < . . . < w s where s = r + t . In other words, w i is the point i ∈ R colored with the color c i from X .For each archetypal arc a not contained in AC there exists a unique 1 ≤ q ≤ s such that a meets w q but not w , . . . , w q − . In this case, define h ( a ) = − q . Then h is a Morse function building up AC from AC . So let a be such an archetypal arc.Let X ′ be the color word obtained from X by removing all colors corresponding tothe nodes contained in a and to the nodes w , . . . , w q − . Then the descending link lk ↓ ( a ) is isomorphic to AC ( C, A ; X ′ ) and by induction, it is ν κ ( lX ′ )-connected.We can estimate lX ′ = lX − l a − ( q − lX − l a − q + 1 ≥ lX − m a − q + 1 ≥ lX − m a − ( m a + m r − 1) + 1= lX − m a − m r + 2Thus, lk ↓ ( a ) is ν κ ( lX − m a − m r + 2)-connected. Consequently, by the Morsemethod, the connectivity of the pair ( AC , AC ) is ν κ ( lX − m a − m r + 2) + 1 = ν κ ( lX )because of κ = 2 m a + m r − d = 3, one can show that the inclusion ι : AC → AC inducesthe trivial map in π m for m ≤ ν κ ( lX ). This proves the theorem in the case d = 1. Remark . The method used in the proof of [14, Proposition 4.11] yields thebetter connectivity ν κ ( lX ) with κ = m a + m r − d = 1.4.2. A contractible complex. From now on, let O be an operad as in Theorem4.3. Furthermore, let X be an object in S := S ( O ). By abuse of notation, theconnected component of S containing the object X will again be denoted by S .Furthermore, we abbreviate Γ := π ( O , X ).As already noted above, the strategy to prove Theorem 4.3 is to apply Brown’scriterion 4.6 to a suitable contractible complex on which the group in question acts.Consider the universal covering category U := U X ( S ) of S based at X . We claim: U is a generalized poset and contractible.This follows from Proposition 2.13. The first claim together with the remarks afterDefinition 2.8 implies that we can form the quotient category U / G where G is thesubgroupoid of U consisting of the transformations in S (lifted to U ) and that U / G is a poset, the underlying poset of U . Recall that Γ acts on U which is encodedin a functor Γ → CAT sending the unique object of Γ to U . One can easily seethat this functor induces a functor Γ → CAT sending the unique object to U / G . Inother words, Γ also acts on U / G . More concretely, an arrow f : X → X in Γ actson an object [ g : X → Y ] of U / G from the right via [ g ] · f := [ f − g ]. This will bethe action to which we want to apply Brown’s criterion. Since U / G is homotopyequivalent to U , the second claim implies that also U / G is contractible. So the firstcondition in Brown’s criterion is satisfied.4.3. Isotropy groups. We continue to verify the conditions in Brown’s criterionfor the action of Γ on U / G . In this subsection, we show that cell stabilizers are oftype F ∞ . First, we note that U / G is indeed a Γ-CW-complex. This follows fromthe following general remark. Remark . Let G be a group acting on a category C . An element g ∈ G fixing acell setwise already fixes its vertices and so fixes the cell pointwise. Consequently, a category with an action of a discrete group G is a G -CW-complex. If C is ageneralized poset, a cell stabilizer is equal to the intersection of the vertex stabilizersof that cell.In the following, we abbreviate T := T ( O ) and I := I ( O ). Lemma 4.12. The groupoid T formed by the transformations in S is of type F ∞ .Proof. By assumption, the groupoid I formed by the degree 1 operations is of type F ∞ . The groupoid T is Mon ( I ) in the planar case, Sym ( I ) in the symmetric caseand Braid ( I ) in the braided case (see Subsection 2.8 for the definitions of thesecategories).Choose a color in each component of I . Let Y be an object in T . We have toshow that Aut T ( Y ) is of type F ∞ . We can assume without loss of generality that Y decomposes as a tensor product of chosen colors: Y = c ⊗ . . . ⊗ c k . In the planarcase we have Aut Mon ( I ) ( Y ) = Aut I ( c ) × . . . × Aut I ( c k )and the claim follows because the Aut I ( c i ) are of type F ∞ . For the symmetric andbraided case, first assume that all the colors c i are equal to one chosen color c . Inthe symmetric case, we then haveAut Sym ( I ) ( Y ) = S k ⋉ Aut I ( c ) k where S k , the symmetric group on k strands, acts by permutation of the factors.More precisely, we have the group homomorphism ϕ : S k → Aut( G k ) σ (cid:2) ( g , . . . , g k ) ( g ⊲σ − , . . . , g k⊲σ − ) (cid:3) which gives a right action of S k on G k by the definition g · σ = g⊲ ( σ⊲ϕ ). Themultiplication in the semidirect product S k ⋉ G k is then given by( σ, g ) ∗ ( σ ′ , g ′ ) := (cid:0) σ ∗ σ ′ , ( g · σ ′ ) ∗ g ′ (cid:1) Since S k is a finite group, it is also of type F ∞ . Since semidirect products of type F ∞ groups are of type F ∞ [23, Exercise 1 on page 176 and Proposition 7.2.2], itfollows that Aut Sym ( I ) ( Y ) is of type F ∞ . In the braided case we haveAut Braid ( I ) ( Y ) = B k ⋉ Aut I ( c ) k where B k , the braid group on k strands, acts via permutation of the factors throughthe projection B k → S k . The braid groups B k are of type F ∞ [38, Theorem A]. Asabove, it follows that Aut Braid ( I ) ( Y ) is of type F ∞ .Remains to handle the case where not all the colors c i lie in the same componentof I . Denote by B ′ k the finite index subgroup of B k consisting of the elements σ with the property c i⊲σ = c i . Since different c i are not connected by an isomorphismin I , we now haveAut Braid ( I ) ( Y ) = B ′ k ⋉ (cid:0) Aut I ( c ) × .. × Aut I ( c k ) (cid:1) where B ′ k still acts by permuting the factors. This action is well-defined due tothe definition of B ′ k . Recall that a group is of type F ∞ if and only if a finiteindex subgroup is of type F ∞ [23, Corollary 7.2.4]. It follows that B ′ k and thusAut Braid ( I ) ( Y ) is of type F ∞ . The symmetric case can be treated similarly. (cid:3) Lemma 4.13. Let P be an object in U / G . Then the stabilizer subgroup Stab Γ ( P ) is of type F ∞ .Proof. Fix an arrow p : X → Y in π ( S ) which represents the object P in U / G ,i.e. [ p ] = P . Let γ ∈ Γ fix the point P . This means [ γ − p ] = [ p ] · γ = [ p ]. Itfollows that there is some transformation t : Y → Y such that γ − p = pt . Thisis equivalent to p − γp = t − which implies that p − γp is an element in Aut T ( Y ). PERAD GROUPS AND THEIR FINITENESS PROPERTIES 41 Conversely, for τ a transformation in Aut T ( Y ), the element pτ p − is contained inStab Γ ( P ). Thus, the mapStab Γ ( P ) → Aut T ( Y ) γ p − γp is an isomorphism with inverse given by τ pτ p − . Since Aut T ( Y ) is of type F ∞ by the previous lemma, the claim follows. Note that this isomorphism depends onthe choice of p . However, two such choices differ by a transformation τ and the twocorresponding isomorphisms differ by conjugation with τ . (cid:3) We say that two operations θ and θ are two-sided transformation equivalent ifthere are transformations α, γ such that θ = α ∗ θ ∗ γ . Proposition 4.14. The stabilizer subgroups of cells in U / G are of type F ∞ .Proof. In the following, we restrict ourselves to the braided case. The planar andsymmetric cases are similar and simpler.We first choose a color in each connected component of I . Next, we choose anoperation in each two-sided transformation class such that the output of the chosenoperation is a chosen color.A non-degenerate cell in the geometric realization of U / G is a sequence of com-posable non-trivial arrows in U / GP ǫ / / P ǫ / / · · · ǫ k − / / P k Let p k : X → Y k be a representing path of P k such that Y k = c ⊗ . . . ⊗ c l is a tensorproduct of chosen colors. In the proofs of Lemmas 4.12 and 4.13, we have seen that p k induces an isomorphism ϕ : Stab Γ ( P k ) → B ′ l ⋉ (cid:0) Aut I ( c ) × .. × Aut I ( c l ) (cid:1) Choose some P i =: P different from P k and observe the arrow ǫ : P → P k whichis the composition of the ǫ j in between. Choose a representing path p : X → Y of P . Then there is exactly one arrow e : Y → Y k representing ǫ . One can compose p and p k with transformations η and λ such that λ : Y k → Y k is a tensor product ofdegree 1 operations λ i : c i → c i and e is a tensor product of chosen operations. Write p ′ k = p k λ for the new representative of P k . To p ′ k corresponds another isomorphism ϕ ′ : Stab Γ ( P k ) → B ′ l ⋉ (cid:0) Aut I ( c ) × .. × Aut I ( c l ) (cid:1) which differs from ϕ by conjugation with λ . Denote the new representative pη of P again by p .Now let γ ∈ Stab Γ ( P k ). Then γ fixes also P , i.e. P · γ = P , if and only if[ p ′ k e − ] = [ p ] = [ p ] · γ = [ γ − p ]= [ γ − p ′ k e − ]= [ p ′ k p ′ k − γ − p ′ k e − ]= [ p ′ k t − γ e − ]where we have set t γ := p ′ k − γp ′ k , an element in the image of the isomorphism ϕ ′ . Therefore, we have to identify all such t γ which satisfy this equation. In otherwords, we look for all t γ such that there is a transformation τ with et γ = τ e Roughly speaking, we look for all t γ which can be pulled through e from thecodomain to the domain. For better readability, we assume without loss of generality that the colors c i are all equal to one color c . In particular, the codomains of ϕ and ϕ ′ are ofthe form B l ⋉ Aut I ( c ) l . Then write e = θ ⊗ . . . ⊗ θ l where the θ i are chosenoperations with codomain the chosen color c . Define H i to be the subgroup ofAut I ( c ) consisting of elements h which can be pulled through the operation θ i ,i.e. there exists a transformation τ with θ i h = τ θ i . Furthermore, let B ∗ l be thefinite index subgroup of B l consisting of the elements σ with the property θ i⊲σ = θ i .Denote by Stab Γ ( P , P k ) the subgroup of Stab Γ ( P k ) which also fixes P . Then theisomorphism ϕ ′ restricts to an isomorphism ϕ ′P : Stab Γ ( P , P k ) → B ∗ l ⋉ ( H × . . . × H l ) =: Λ P where the subgroup B ∗ l still acts via permutation of the factors and this is well-defined due to the definition of B ∗ l . The proof of this is straightforward and usesthe fact that two two-sided transformation equivalent θ i must be equal.Recall that ϕ ′ differs from ϕ by conjugation with λ . So the image of Stab Γ ( P , P k )under ϕ is its image under ϕ ′ conjugated with λ . More precisely, ϕ restricts to anisomorphism ϕ P : Stab Γ ( P , P k ) → λ Λ P λ − =: Ω P Consider the pure braid group P l which is a finite index subgroup of B l . It is alsoa finite index subgroup of B ∗ l . Recall that we have λ = λ ⊗ . . . ⊗ λ l where the λ i are degree 1 operations. We have λ (cid:0) P l ⋉ ( H × . . . × H l ) (cid:1) λ − = λ (cid:0) P l × ( H × . . . × H l ) (cid:1) λ − = P l × (cid:0) λ H λ − × . . . × λ l H l λ − l (cid:1) = P l × (cid:0) H P × . . . × H P l (cid:1) where H P i := λ i H i λ − i is isomorphic to H i . This is a finite index subgroup of Ω P .Remains to consider the case when γ ∈ Stab Γ ( P k ) fixes more than one additionalvertex P i . For this we have to show that the intersectionΩ P ∩ . . . ∩ Ω P k − ⊂ B l ⋉ Aut I ( c ) l is of type F ∞ . For better readability, we assume without loss of generality that k = 2. Then the last statement is equivalent to (cid:16) P l × (cid:0) H P × . . . × H P l (cid:1)(cid:17) ∩ (cid:16) P l × (cid:0) H P × . . . × H P l (cid:1)(cid:17) = P l × (cid:16)(cid:0) H P ∩ H P (cid:1) × . . . × (cid:0) H P l ∩ H P l (cid:1)(cid:17) being of type F ∞ since it is a finite index subgroup. This is true because P l is oftype F ∞ and all the groups H P i ∩ H P i are of type F ∞ . The latter statement istrue because the groups H P i ∩ H P i are subgroups of Aut I ( c ) which is of type F + ∞ .This completes the proof of the proposition. (cid:3) Finite type filtration. To apply Brown’s criterion to the Γ-CW-complex U / G , we need a filtration by Γ-CW-subcomplexes ( U / G ) n which are of finite type.Observe that the degree function on S induces degree functions on U and U / G .Define S n resp. U n resp. ( U / G ) n to be the full subcategories spanned by the objectsof degree at most n . Note that we have U n / G n = ( U / G ) n where G n = G ∩ U n . Inthe following, we want to show that ( U / G ) n only has finitely many Γ-equivariantcells in each dimension.Choose one operation in each very elementary transformation class and denotethe resulting set of operations by S . By the assumptions in Theorem 4.3, S is afinite set. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 43 Observation . Let θ ∈ S and γ be a degree 1 operation such that θ ∗ γ is defined.Then, by Proposition 3.19, θ ∗ γ is again very elementary and there is a θ ′ ∈ S anda transformation τ with θ ∗ γ = τ ∗ θ ′ .Denote by Ω the set of all identity operations together with all operations ofdegree at most n which are obtained by partially composing operations in S . Notethat Ω is finite because S is finite and there are only finitely many colors by as-sumption. Denote by Λ the set of arrows in S n which are obtained by taking tensorproducts of operations in Ω. Again, the set Λ is finite.Let Λ ∗ p ⊂ Λ p be the subset of p -tuples of composable arrows in Λ. We claimthat there is a surjective functionΛ ∗ p ։ (cid:8) p -cells in ( U / G ) n (cid:9)(cid:14) Γwhich proves that there are only finitely many Γ-equivariant cells in ( U / G ) n . Let( e , . . . , e p − ) ∈ Λ ∗ p . Choose a path p : X → dom( e ). Define paths p k : X → dom( e k ) by the composite p k := p e . . . e k − . The p i represent objects P i and the e i represent arrows ǫ i : P i → P i +1 in ( U / G ) n . Thus, the sequence ǫ , . . . , ǫ p − givesa p -cell in ( U / G ) n . This p -cell surely depends on the choice of p but two suchchoices give equivalent p -cells modulo the action of Γ. So we get a well-definedfunction as above.Remains to show that this function is indeed surjective. Consider a p -cell in( U / G ) n in the form of a string P ǫ −→ P ǫ −→ · · · ǫ p − −−−→ P p of composable arrows in ( U / G ) n . For each P i we can choose representatives P i in U n . Then each ǫ i is represented by a unique arrow e i : P i → P i +1 in U n . We nowwant to change these representatives so that each e i lies in Λ.Start with the last arrow e p − = [ σ, Θ]. Let T be the set of operations of theform τ ∗ θ where τ is a transformation and θ ∈ S . In other words, T is the set of allvery elementary operations. Each higher degree operation θ in the sequence Θ canbe written, up to transformation, as a partial composition of operations in T (seethe remarks after Definition 3.16). It follows θ = s ∗ ψ where s is a transformationand ψ is an operation decomposable into operations of the form ( γ , . . . , γ k ) ∗ ξ with ξ ∈ S and γ i of degree 1. Using Observation 4.15, we can pull the degree 1operations to the domain of ψ , starting with the rightmost degree 1 operations, andobtain θ = s ∗ ψ where s is a transformation and ψ is an operation decomposableinto operations of S . We now have e p − = τ ∗ Ψ where τ is some transformation andΨ is simply a tensor product of identities or higher degree operations decomposableinto operations of S . By changing the representives P p − , e p − and e p − in theirrespective classes modulo the subgroupoid G , we can assume τ = id and thus that e p − lies in Λ. We can now repeat this argument with e p − and then with e p − andso forth until we have changed each e i to lie in Λ. This proves surjectivity.4.5. Connectivity of the filtration. It remains to show the connectivity state-ment in Brown’s criterion, i.e. we have to show that the connectivity of the pair (cid:0) ( U / G ) n , ( U / G ) n − (cid:1) tends to infinity as n → ∞ . To show this, we apply the Morsemethod for categories. The degree function on U / G is a Morse function and thecorresponding filtration is exactly ( U / G ) n . Thus, we have to prove that the con-nectivity of the descending link lk ↓ ( K ) tends to infinity as the degree of the object K tends to infinity. Note that the descending up link lk ↓ ( K ) is always empty, sowe have lk ↓ ( K ) = lk ↓ ( K ). Definition 4.16. An arrow [ σ, Θ] in S is called (very) elementary if it is not atransformation and every higher degree operation in Θ is (very) elementary. An arrow in U is called (very) elementary if the corresponding arrow in S is (very) ele-mentary. An arrow in U / G is called (very) elementary if there is a (very) elementaryrepresentative in U .It follows from Proposition 3.19 that the number of (very) elementary operationsin an arrow a ∈ U does does not change if we replace a by another representativein the class [ a ] ∈ U / G . In particular, if the arrow α ∈ U / G is (very) elementary,then all representing arrows a ∈ U of α are (very) elementary.The data of an object in lk ↓ ( K ) consists of an object Y in U / G with deg( Y ) < deg( K ) and an arrow α : K → Y in U / G . Now we define Core( K ) to be the fullsubcategory of lk ↓ ( K ) spanned by the objects ( Y , α ) where α is a very elementaryarrow. Denote by Corona( K ) the full subcategory of lk ↓ ( K ) spanned by the objects( Y , α ) with α an elementary arrow. So we haveCore( K ) ⊂ Corona( K ) ⊂ lk ↓ ( K )and we will study the connectivity of these spaces successively.4.5.1. The core. In this subsubsection, we adopt the normal form point of view ofSubsection 3.2: Arrows in S are always represented by a unique pair ( σ, Θ) suchthat σ − is unpermuted resp. unbraided on the domains of the operations in thesequence Θ.We say that two operations θ and θ are right transformation equivalent if thereis a transformation γ such that θ = θ ∗ γ . Recall from Proposition 3.19 that beingelementary or very elementary is invariant under right transformations.The object K in U / G is a class of objects in U modulo transformations. Fixsome representing object K . Then the objects in Core( K ) are in one to one cor-respondence with pairs ( Y, a ) where Y is an object in U with deg( Y ) < deg( K )and a : K → Y is a very elementary arrow in U modulo transformations on thecodomain (compare with Remark 2.6). Choose one operation in each right trans-formation equivalence class and denote the resulting set of operations by R . Wechoose the identity for a class of degree 1 operations so that the degree 1 operationsin R are identities. Now define a very elementary R -arrow to be a very elementaryarrow ( σ, Θ) in S such that the operations in Θ are elements of R . Thus, Θ is atensor product of identities and at least one very elementary operation lying in R .This notion of very elementary R -arrows can be lifted to arrows in U . Now theobjects in Core( K ) are in one to one correspondence with pairs ( Y, a ) where Y isan object with deg( Y ) < deg( K ) and a : K → Y is • (planar case) a very elementary R -arrow. • (symmetric case) a very elementary R -arrow modulo colored permutationson the codomain. • (braided case) a very elementary R -arrow modulo colored braidings on thecodomain.The equivalence relation modulo braidings on the codomain is called “dangling” in[9] because these objects may be visualized as a braiding where some strands atone end are connected by very elementary operations in R , called “feet”, and theseare allowed to dangle freely (see [9, Figure 9]).Now let C be the set of colors of the operad O . We define a set of archetypes A as follows: For each operation in R , form an archetype with identifier this operationand with color word the domain of that operation. The object K in U is a path ofarrows in S modulo homotopy. It starts at the color word X and ends at some othercolor word T . Consider the simplicial complex AC d ( C, A ; T ) from Subsection 4.1.It can be seen as a poset of simplices with an arrow from a simplex σ to anothersimplex σ ′ if and only if σ is a face of σ ′ . PERAD GROUPS AND THEIR FINITENESS PROPERTIES 45 Proposition 4.17. The category Core( K ) is a poset and isomorphic, as a poset, to AC d ( C, A ; T ) where d = 1 in the planar case, d = 2 in the braided case and d = 3 in the symmetric case.Proof. We restrict our attention to the braided case, i.e. d = 2. The other two casesare much simpler.First, it is clear that Core( K ) is a poset since U / G is a poset. We want to under-stand the poset structure a bit better: Let Λ be an object of Core( K ) in the formof a very elementary R -arrow K → Y modulo dangling. Fix some very elementary R -arrow λ representing this class with the property that the colored braiding ofthat arrow is unbraided not only on the sets of strands connected to single opera-tions but also on the set of strands connected to identity operations. Then arrowsin Core( K ) with domain Λ are in one to one correspondence with very elementary R -arrows α in U , modulo dangling, such that the very elementary operations of α only connect to identity operations of λ in the composition λ ∗ α (since com-positions of very elementary operations are not very elementary anymore). Thefollowing diagram, in which the gray triangles are identity operations, illustratessuch a situation: λ α These considerations yield the following interpretation of the poset structure: Wehave Λ → Λ ′ if and only if there is a very elementary R -arrow λ representing thedangling class Λ such that adding very elementary operations of R to loose strandsof λ (i.e. strands connected to identity operations) gives a very elementary R -arrowrepresenting the dangling class Λ ′ .We will consider an isomorphism of posetscomb : Core( K ) → AC ( C, A ; T )called “combing” as in [9, Section 4] and its inverseweave: AC ( C, A ; T ) → Core( K )which we call “weaving”.To define the first map, start with an object Λ in Core( K ). As above, it is a veryelementary R -arrow in normal form modulo dangling. Thus, it is represented by acolored braid with unbraided strands connected by very elementary operations in R . Think of the domain of the braid as being fixed on the line L := { ( x, , | x ∈ R } ⊂ R the codomain as being fixed on the line L := { ( x, , | x ∈ R } ⊂ R and visualize the operations as straight lines in L connecting the ends of thecorresponding strands. Now “combing straight” the braid means moving aroundthe ends of the braid in the plane P := R × { } ⊂ R such that the whole braidbecomes unbraided. The segments representing the operations get deformed in P this way and in fact become the archetypal arcs in comb(Λ). They are admissible because the braid was required to be unbraided on the domains of the operations.This process is visualized in [9, Figure 17]. Note that combing does not dependon the representative under dangling, so it is a well-defined map on the objects ofCore( K ). It also respects the poset structures, so it is a map of posets.To define the second map, start with an archetypal arc system A . This is a prioriembedded in R but embed it in R via the embedding R × { } ⊂ R . Connectthe nodes of the archetypal arc system with the line L by straight lines parallelto the third coordinate axis. The process of weaving first tries to separate thearchetypal arcs by moving the nodes in the plane P . Here, being separate meansbeing separated by a straight line in P parallel to the second coordinate axis. Also,the set of nodes which are not contained in an arc should be separated from thearcs. By doing these moves, the vertical strands connecting the nodes with the line L become braided in a certain way. The separation process is always possible butthe resulting braid is not unique (think of two nodes connected by an arc and turnaround the arc several times). To make the resulting braid unique (up to dangling),we additionally require that the subbraid determined by an archetypal arc neverbecomes braided during the separation process. This can be achieved for exampleby the following additional movement rule: The nodes of an archetypal arc alwayshave to stay on the same line L in P parallel to the first coordinate axis. This line L may move up and down and the nodes of the archetypal arc may move left and righton L but they must never cross each other on L . Then, when the archetypal arcsare separated from each other and from the isolated nodes, the property admissibleof the archetypal arcs ensures that they can be homotoped to straight lines lyingin L . The following figure visualizes this process:Replacing the archetypal arcs by the identifier operations of the correspondingarchetype yields a representative of weave( A ) and the class modulo dangling doesnot depend on the weaving process. Thus, we get a well-defined map on the objectsof AC ( C, A ; T ). It also respects the poset structures, so it is a map of posets. (cid:3) It follows from the finiteness of V E that the set of archetypes A is of finite type(though not finite in general) and from the color-tameness of O that it is tame.More precisely, let m V be the largest degree of very elementary classes and m C be the smallest natural number greater than the degree of any reduced object in S . Then we can set m a = m V and m r = m C in Theorem 4.9. We thus get thefollowing Corollary 4.18. Core( K ) is ν d (deg K ) -connected where ν ( l ) := (cid:22) l − m C m V + m C − (cid:23) − PERAD GROUPS AND THEIR FINITENESS PROPERTIES 47 ν ( l ) := (cid:22) l − m C m V − (cid:23) − ν ( l ) := (cid:22) l − m C m V − (cid:23) − Here, d = 1 corresponds to the planar case, d = 2 to the braided case and d = 3 tothe symmetric case. The corona. We build up Corona( K ) from Core( K ) using again the Morsemethod for categories. We then get a connectivity result for the corona from theconnectivity result for the core. The idea is attributed to [20].We assumed O to be of finite type, i.e. the set of elementary classes E is finite.Let m E be the largest degree of elementary classes. An object in Corona( K ) is apair ( Y , α : K → Y ) where deg( Y ) < deg( K ) and α is an elementary arrow in U / G .For 2 ≤ k ≤ m E denote by kse ( α ) the number of strictly elementary operations ofdegree k in any representative of α . Define f (cid:0) ( Y , α ) (cid:1) := (cid:0) m E se ( α ) , m E − se ( α ) , . . . , se ( α ) , deg( Y ) (cid:1) Order the values of f lexicographically. Then f becomes a Morse function buildingup Corona( K ) from Core( K ). Define µ ( l ) := (cid:22) l − m C m V + m C + m E (cid:23) − µ ( l ) := (cid:22) l − m C m V + m E (cid:23) − µ ( l ) := (cid:22) l − m C m V + m E (cid:23) − Proposition 4.19. For each object ( Y , α ) in Corona( K ) which is not an object in Core( K ) , the descending link lk ↓ ( Y , α ) with respect to the Morse function f aboveis µ d (deg K ) -connected. From Theorem 2.20 we get that Core( K ) and Corona( K ) share the same ho-motopy groups up to dimension µ d (deg K ). We already know that Core( K ) is ν d (deg K )-connected. Furthermore, we have ν d ( l ) ≥ µ d ( l ). Consequently, we getthe following Corollary 4.20. Corona( K ) is µ d (deg K ) -connected. In particular, its connectivitytends to infinity as deg( K ) → ∞ . In the rest of this subsubsection, we give a proof of Proposition 4.19. We dis-tinguish between two sorts of objects ( Y , α ) in Corona( K ) which are not objectsin Core( K ): Such an object is called mixed if there is at least one very elementaryoperation in α . It is called pure if there is no very elementary operation in α . Lemma 4.21. Let ( Y , α ) be mixed. Then lk ↓ ( Y , α ) and therefore lk ↓ ( Y , α ) iscontractible. In particular, Proposition 4.19 is true for mixed objects.Proof. The data of an object in lk ↓ ( Y , α ) is Ω = (cid:0) ( L , β ) , β (cid:1) where L is an object in U / G , β is an elementary arrow in U / G , β is an arrow in U / G such that β β = α and ( L , β ) forms an object in Corona( K ) of strictly smaller Morse height than( Y , α ). Let Ω ′ = (cid:0) ( L ′ , β ′ ) , β ′ (cid:1) be another such object. An arrow Ω → Ω ′ is represented by an arrow δ : L → L ′ such that β δ = β ′ and δβ ′ = β . K α / / β (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄ β ′ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ YL β ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ δ / / L ′ β ′ ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ It follows that lk ↓ ( Y , α ) is a poset since U / G is a poset.Choose representatives K and Y of K and Y . Then α is represented by a uniquearrow a : K → Y . We can choose K such that a is a tensor product of higherdegree operations and identities. Let a v : K → Y v be the arrow obtained from a by replacing all strictly elementary operations θ with deg( θ ) identity operations.Let a se : Y v → Y be the arrow obtained from a by replacing all very elementaryoperations by one identity operation each. We have a v a se = a . An example of a, a v , a se is pictured below. There, a white triangle is a placeholder for a strictlyelementary operation. A black triangle indicates a very elementary operation. Astraight horizontal line represents an identity operation. a a v a se Set Y v := [ Y v ] and α v := [ a v ] as well as α se := [ a se ]. Then ( Y v , α v ) is an object inCore( K ) and α se represents an arrow ( Y v , α v ) → ( Y , α ) in Corona( K ). Moreover,the pair Ξ := (cid:0) ( Y v , α v ) , α se (cid:1) is an object in lk ↓ ( Y , α ).Let Ω = (cid:0) ( L , β ) , β (cid:1) be an object in lk ↓ ( Y , α ). We define another object F (Ω) = (cid:0) ( M , γ ) , γ (cid:1) of lk ↓ ( Y , α ) as follows: Choose a representative L of L such that β is represented by b : L → Y which is a tensor product of identities and higherdegree operations. Then β is represented by a unique b : K → L . Note that b b = a . Think of b as splitting higher degree operations of a into operations ofsmaller degree and of b as merging them back to their original form. Now definethe arrows g : K → M and g : M → Y to be the same splitting of a with the onlyexception that no very elementary operation of a is splitted. An example fittingto the example above is pictured below. There, a gray triangle is a placeholderfor an elementary operation or a degree 1 operation, a blue triangle can be anyoperation and a dot on a straight horizontal line indicates a possibly non-trivialdegree 1 operation. b b g g Now set M := [ M ] and γ := [ g ] as well as γ = [ g ]. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 49 It is not hard to see that Ω F (Ω) extends to a functor lk ↓ ( Y , α ) → lk ↓ ( Y , α )which means that whenever we have an arrow δ : Ω → Ω ′ , then there is an arrow∆ : F (Ω) → F (Ω ′ ). M ′ γ ′ ●●●●●●●●●●●●●●●●●●●● M γ ( ( PPPPPPPPPPPPPPPPPP ∆ O O K γ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ γ ′ : : ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ α / / α v ❅❅❅❅❅❅❅❅❅❅ β ′ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ β ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ YY v α se ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ L ′ β ′ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧ L β E E ☛☛☛☛☛☛☛☛☛☛☛☛☛☛ δ O O We also have arrows ξ Ω : Ξ → F (Ω) and ι Ω : Ω → F (Ω). M γ ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ K α / / α v ❆❆❆❆❆❆❆❆❆ β * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ γ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ YY v α se ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ξ Ω F F ✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌ L β ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧ ι Ω W W ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ The arrow ξ Ω is represented by an arrow x Ω : Y v → M which satisfies a v x Ω = g and x Ω g = a se . The arrow ι Ω is represented by i Ω : L → M which satisfies b i Ω = g and i Ω g = b . In the example from above, these arrows look as follows: x Ω i Ω The claim of the proposition now follows from item iii) in Subsection 2.5 appliedto the functor F and the object Ξ. (cid:3) Lemma 4.22. Let ( Y , α ) be pure. Then lk ↓ ( Y , α ) is µ d (deg K ) -connected andProposition 4.19 is true for pure objects.Proof. Choose representatives K and Y of K and Y such that a : K → Y repre-senting α is a tensor product of higher degree operations and identities.First observe the descending up link lk ↓ ( Y , α ). An object in lk ↓ ( Y , α ) is a pair (cid:0) ( L , β ) , β (cid:1) with f ( L , β ) < f ( Y , α ) and β β = α . When choosing a representa-tive L of L , we get unique representatives b : K → L of β and b : L → Y of β such that b b = a . As in the proof of the previous lemma, b can be interpretedas splitting higher degree operations of a into operations of smaller degree and b as merging them back to their original form. Denote by A i the full subcategory of lk ↓ ( Y , α ) spanned by the objects which only split the i ’th higher degree operation in a . Denote by n the number of higher degree operations in a . Observe now thatwhen splitting operations in a one by one, then we can also split all that operationsat once. This observation reveals that lk ↓ ( Y , α ) = A ◦ . . . ◦ A n is the Grothendieck join of the A i (see Subsection 2.9). Note that the categories A i are all non-empty since all the higher degree operations in a are elementary butnot very elementary and splitting such a strictly elementary operation decreasesthe Morse height. Thus, lk ↓ ( Y , α ) is ( n − lk ↓ ( Y , α ). Objects are pairs (cid:0) ( L , β ) , β (cid:1) with f ( L , β ) < f ( Y , α ) and αβ = β . When choosing a representative L of L ,we get unique representatives b : K → L of β and b : Y → L of β such that ab = b . Looking at the Morse function f for the corona, one sees that the higherdegree operations of b must be very elementary operations which only composewith identity operations of a . At this point, we have to distinguish between theplanar case on the one hand and the braided resp. symmetric case on the other.We start with the braided resp. symmetric case: The arguments in the proof ofProposition 4.17 reveal that lk ↓ ( Y , α ) is isomorphic to AC d ( C, A ; T ′ ) where T ′ isthe color word obtained from the codomain of a (viewed as an arrow in S ) afterdeleting the higher degree operations. Denote by l the length of T ′ , i.e. the numberof identity operations in a . Then we already know that AC d ( C, A ; T ′ ) is ν d ( l )-connected (compare with Corollary 4.18). Consequently, the connectivity of thedescending link lk ↓ ( Y , α ) = lk ↓ ( Y , α ) ∗ lk ↓ ( Y , α ) is n + ν d ( l ) = n + (cid:22) l − m C m V − (cid:23) − ≥ n + (cid:22) deg K − n m E − m C m V − (cid:23) − ≥ n + (cid:22) deg K − n m E − m C m V + m E (cid:23) − (cid:22) deg K − m C + 2 m V n m V + m E (cid:23) − ≥ (cid:22) deg K − m C m V + m E (cid:23) − µ d (deg K )where we have used that n m E + l ≥ deg K .Now we turn to the planar case: An identity component in a is a maximalsubsequence of identity operations. Let m be the number of identity componentsand denote by l i for i = 1 , . . . , m the length of the i ’th identity component. Denoteby l the total number of identity operations in a , i.e. the sum of the l i . Define B i to be the full subcategory of lk ↓ ( Y , α ) spanned by the objects which only add veryelementary operations into the i ’th identity component. Observe now that whenadding very elementary operations into different identity components one by one,then we can also add all that operations at once. This reveals that lk ↓ ( Y , α ) is theGrothendieck join of the B i . Note, though, when inspecting the direction of thearrows in lk ↓ ( Y , α ), one sees that it is in fact the dual Grothendieck join. So wehave lk ↓ ( Y , α ) = B • . . . • B m Similarly as in the braided resp. symmetric case, B i is isomorphic to AC ( C, A ; T i )where T i is the color word obtained from the codomain of a after deleting alloperations except the identity operations of the i ’th identity component. The length PERAD GROUPS AND THEIR FINITENESS PROPERTIES 51 of T i is l i . Then we already know that AC ( C, A ; T i ) is ν ( l i )-connected. Therefore,the connectivity of lk ↓ ( Y , α ) is at least2 m − m X j =1 ν ( l j )Thus, the connectivity of lk ↓ ( Y , α ) is at least n + 2 m − m X j =1 ν ( l j ) ≥ n + m − m X j =1 (cid:22) l j − m C m V + m C (cid:23) ≥ n − m X j =1 l j − m C m V + m C = n − l − m m C m V + m C ≥ n − K − n m E − ( n + 1) m C m V + m C ≥ n + deg K − n m E − n m C − m C m V + m C + m E − 2= deg K − m C + 2 m V n m V + m C + m E − ≥ deg K − m C m V + m C + m E − ≥ µ (deg K )where we have used in the fourth step that m ≤ n + 1 and n m E + l ≥ deg K . (cid:3) The whole link. In this last step, we show that the inclusionCorona( K ) ⊂ lk ↓ ( K )is a homotopy equivalence. It then follows from Corollary 4.20 that the connectivityof lk ↓ ( K ) tends to infinity as deg( K ) → ∞ which is what we wanted to show inorder to finish the proof of Theorem 4.3. This step is analogous to the reduction tothe Stein space of elementary intervals in [39]. We again apply the Morse methodfor categories to build lk ↓ ( K ) up from Corona( K ). The Morse function on objectsof lk ↓ ( K ) which do not lie in Corona( K ) is given by f (cid:0) ( Y , α ) (cid:1) := − deg( Y )We have lk ↓ ( Y , α ) = ∅ and thus lk ↓ ( Y , α ) = lk ↓ ( Y , α ) with respect to this Morsefunction. Similarly as in the proofs of Lemmas 4.21 and 4.22, we obtain lk ↓ ( Y , α ) = A ◦ . . . ◦ A n where the A i are full subcategories of lk ↓ ( Y , α ) spanned by the objects which corre-spond to splitting exactly one of the n higher degree operations in a representative a of α . At least one of these operations must be non-elementary since ( Y , α ) is notan object in Corona( K ). Without loss of generality, assume that A corresponds tosuch a non-elementary higher degree operation. If we show that A is contractible,it follows that lk ↓ ( Y , α ) is contractible. Thus, we are building lk ↓ ( K ) up fromCorona( K ) along contractible descending links and it follows from Theorem 2.20that the inclusion Corona( K ) ⊂ lk ↓ ( K ) is a homotopy equivalence. That A iscontractible follows from Proposition 4.24 below.First, we want to reinterprete the defining property of E as the spine of T C ∗ ( O )in terms of the category U / G . Lemma 4.23. Let α : K → Y be a non-elementary arrow in U / G such that deg( K ) = n > and deg( Y ) = 1 . Then there is a unique pair ( α , α ) of arrows in U / G (calledthe maximal elementary factorization of α ) such that α is elementary, α α = α and such that the following universal property is satisfied: Whenever ( β , β ) isanother pair with β elementary and β β = α (called an elementary factorizationof α ), then there is a unique arrow γ with α γ = β and γβ = α . K α (cid:15) (cid:15) α (cid:6) (cid:6) ✌✌✌✌✌✌✌✌✌✌✌✌✌ β (cid:24) (cid:24) ✶✶✶✶✶✶✶✶✶✶✶✶✶ Q α / / γ A A ❄ ❏ ❚ ❴ ❥ t ⑧ Y P β o o Proof. Recall that U / G is a poset. So there is at most one such γ . If also ( β , β )satisfies the universal property, then we must have Q = P and consequently α = β as well as α = β . This shows the uniqueness statements.Remains to prove the existence of such a pair: Choose representatives K, Y of K , Y . Then α is represented by a unique arrow a : K → Y . Note that a is just anoperation since deg( Y ) = 1. Denote the transformation class of a by Ω. Its degreeis deg( K ) = n > E as the spine, there is a greatest elementary class Θ with the property Θ < Ω.This implies that there is an operation θ ∈ Θ and an arrow q in S such that q ∗ θ = a in S . Define Q := K ∗ q as an object in U and further Q := [ Q ] as an object in U / G . The arrows q : K → Q resp. θ : Q → Y in U represent arrows α resp. α in U / G such that α α = α and α is elementary.These two arrows satisfy the universal property: Let b : K → P and b : P → Y be representatives of β : K → P and β : P → Y . Obviously, the transformationclass [ b ] of b is elementary and satisfies [ b ] < [ a ] = Ω. Since Θ = [ θ ] is thegreatest such class, we obtain [ b ] ≤ [ θ ]. This means that there is an arrow g in S such that g ∗ b = θ in S . If g is interpreted as an arrow Q → P in U , then itrepresents an arrow γ : Q → Y in U / G which satisfies γβ = α . We then also have α γ = β since U / G is a poset. (cid:3) We now turn to the announced proposition which concludes the proof of themain theorem. Proposition 4.24. Let α : K → Y be a non-elementary arrow in U / G such that deg( K ) = n > and deg( Y ) = 1 . Let M be the full subcategory of K↓ ( U / G ) n − spanned by the objects ( Z , β : K → Z ) with deg( Z ) > and L := M↓ ( Y , α ) the descending up link of ( Y , α ) with respect to the Morse function f above. Then L is contractible.Proof. Note that the data of an object of L is a non-trivial factorization of α , i.e. apair ( α , α ) of arrows in U / G such that α = id = α and α α = α . An arrowfrom ( α , α ) to ( β , β ) is an arrow γ such that α γ = β and γβ = α . Clearly, L is a poset.Apply Lemma 4.23 above to obtain a maximal elementary factorization ( α , α )of α . Note that ( α , α ) is an object of L and the universal property says that thisobject is initial among the objects ( β , β ) of L with β elementary. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 53 More generally, for an object ( ǫ , ǫ ) of L with ǫ non-elementary, we can applythe lemma to obtain a maximal elementary factorization ( ǫ ∗ , ǫ ∗ ) of ǫ . Then define F ( ǫ , ǫ ) := ( ǫ ǫ ∗ , ǫ ∗ ) which is again an object in L . If ǫ is already elementary, weset ǫ ∗ = id and ǫ ∗ = ǫ so that F ( ǫ , ǫ ) = ( ǫ , ǫ ).We claim that F extends to a functor L → L . So let ( ǫ , ǫ ) and ( β , β ) be twoobjects of L and γ : ( ǫ , ǫ ) → ( β , β ) an arrow in L . We have to show that thereis an arrow ϕ : F ( ǫ , ǫ ) → F ( β , β ). • α (cid:15) (cid:15) ǫ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ β (cid:31) (cid:31) ❅❅❅❅❅❅❅ • γ / / ǫ (cid:31) (cid:31) ❅❅❅❅❅❅❅ ǫ ∗ (cid:15) (cid:15) • β (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ β ∗ (cid:15) (cid:15) • ǫ ∗ / / ϕ B B ❂ ❍ ❚ ❴ ❥ ✈ ✁ • • β ∗ o o Observe first that if ǫ ∗ = id, then γβ ∗ is an arrow F ( ǫ , ǫ ) → F ( β , β ) as required.Else, observe that the pair ( γβ ∗ , β ∗ ) is another elementary factorization of ǫ . Thus,by the universal property, we get a unique arrow ϕ such that ϕβ ∗ = ǫ ∗ and ǫ ∗ ϕ = γβ ∗ . This amounts to an arrow F ( ǫ , ǫ ) → F ( β , β ).Since F ( ǫ , ǫ ) is an elementary factorization of α , we get an arrow ( α , α ) → F ( ǫ , ǫ ) for each object ( ǫ , ǫ ) in L . Furthermore, ǫ ∗ clearly gives an arrow( ǫ , ǫ ) → F ( ǫ , ǫ ). The claim of the proposition now follows from item iii) inSubsection 2.5 applied to the functor F and the object ( α , α ). (cid:3) Applications. Suboperads of endomorphism operads. Consider the example with squaresand triangles, the cube cutting operads (planar or symmetric) and the local simi-larity operads from Subsubsection 3.5.2. There, we have seen that they all satisfythe cancellative calculus of fractions. The squares and triangles operad and thecube cutting operads are of finite type. The local similarity operads are of finitetype if and only if there are only finitely many Sim X -equivalence classes of balls,so we will assume this in the following. Then, in all three cases, the groupoid I ( O )is finite.In order to apply Theorem 4.3, it therefore remains to check color-tameness.The cube cutting operads are monochromatic, so color-tameness is trivially satisfiedhere. The squares and triangles operad has two colors (the square and the triangle).It is easy to check that any sequence of at least five squares and triangles is thedomain of a very elementary arrow in S ( O ). Consequently, it is color-tame as well.In general, a local similarity operad is not color-tame.As a special case, we obtain that the higher dimensional Thompson groups nV are of type F ∞ . This has been shown before in [20].The one dimensional cube cutting operads (planar or symmetric) with trivialgroupoid of degree 1 operations yield the groups of piecewise linear homeomor-phisms of the unit (Cantor) interval studied in [39] and from the main theorem, itfollows that they are of type F ∞ . This has already been shown in [39].The finiteness result for the local similarity groups has also been obtained in[16, Theorem 6.5]. The hypothesis in this theorem consists of demanding that thefinite similarity structure posseses only finitely many Sim X -equivalence classes ofballs and of the property rich in simple contractions which is implied by the easierto state property rich in ball contractions [16, Definition 5.12]. It is not hard to see that the latter property exactly means that O , the local similarity operad associatedto Sim X , is color-tame.4.6.2. Ribbon Thompson group. The braided operad O with transformations dis-cussed in Subsubsection 3.5.3 satisfies the cancellative calculus of fractions. It ismonochromatic and therefore color-tame. There is only one very elementary trans-formation class and thus, O is of finite type. The groupoid I ( O ) is the group Z which is of type F + ∞ . The main theorem yields that the Ribbon Thompson group RV is of type F ∞ . References [1] J. B´enabou, Some remarks on -categorical algebra. I , Bull. Soc. Math. Belg. S´er. A (1989), no. 2, 127–194.[2] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups , Invent. Math. (1997), no. 3, 445–470.[3] F. Borceux, Handbook of categorical algebra. 1 , Encyclopedia of Mathematics and its Appli-cations, vol. 50, Cambridge University Press, Cambridge, 1994.[4] M. G. Brin, Higher dimensional Thompson groups , Geom. Dedicata (2004), 163–192.[5] , The algebra of strand splitting. I. A braided version of Thompson’s group V , J.Group Theory (2007), no. 6, 757–788.[6] M. G. Brin and C. C. Squier, Groups of piecewise linear homeomorphisms of the real line ,Invent. Math. (1985), no. 3, 485–498.[7] K. S. Brown, Finiteness properties of groups , Proceedings of the Northwestern conference oncohomology of groups (Evanston, Ill., 1985), 1987, pp. 45–75.[8] K. S. Brown and R. Geoghegan, An infinite-dimensional torsion-free FP ∞ group , Invent.Math. (1984), no. 2, 367–381.[9] K.-U. Bux, M. G. Fluch, M. Marschler, S. Witzel, and M. C. B. Zaremsky, The braidedThompson’s groups are of type F ∞ , arXiv:1210.2931v2 (2014).[10] D.-C. Cisinski, La classe des morphismes de Dwyer n’est pas stable par retractes , CahiersTopologie G´eom. Diff´erentielle Cat´eg. (1999), no. 3, 227–231 (French, with English sum-mary).[11] W. G. Dwyer and D. M. Kan, Calculating simplicial localizations , J. Pure Appl. Algebra (1980), no. 1, 17–35.[12] J. Dydak, A simple proof that pointed FANR-spaces are regular fundamental retracts ofANR’s , Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. (1977), no. 1, 55–62(English, with Russian summary).[13] D. S. Farley, Actions of picture groups on CAT(0) cubical complexes , Geom. Dedicata (2005), 221–242.[14] , Finiteness and CAT(0) properties of diagram groups , Topology (2003), no. 5,1065–1082.[15] , Homological and finiteness properties of picture groups , Trans. Amer. Math. Soc. (2005), no. 9, 3567–3584 (electronic).[16] D. S. Farley and B. Hughes, Finiteness properties of some groups of local similarities , Proc.Edinb. Math. Soc. (2) (2015), no. 2, 379–402.[17] Z. Fiedorowicz, A counterexample to a group completion conjecture of J. C. Moore , Algebr.Geom. Topol. (2002), 33–35 (electronic).[18] M. Fiore and T. Leinster, An abstract characterization of Thompson’s group F , SemigroupForum (2010), no. 2, 325–340.[19] T. M. Fiore, W. L¨uck, and R. Sauer, Euler characteristics of categories and homotopy col-imits , Doc. Math. (2011), 301–354.[20] M. G. Fluch, M. Marschler, S. Witzel, and M. C. B. Zaremsky, The Brin-Thompson groups sV are of type F ∞ , Pacific J. Math. (2013), no. 2, 283–295.[21] P. Freyd and A. Heller, Splitting homotopy idempotents. II , J. Pure Appl. Algebra (1993),no. 1-2, 93–106.[22] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory , Ergebnisse der Math-ematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967.[23] R. Geoghegan, Topological methods in group theory , Graduate Texts in Mathematics, vol. 243,Springer, New York, 2008.[24] P. G. Goerss and J. F. Jardine, Simplicial homotopy theory , Modern Birkh¨auser Classics,Birkh¨auser Verlag, Basel, 2009.[25] V. Guba and M. Sapir, Diagram groups , Mem. Amer. Math. Soc. (1997), no. 620. PERAD GROUPS AND THEIR FINITENESS PROPERTIES 55 [26] B. Hughes, Local similarities and the Haagerup property , Groups Geom. Dyn. (2009), no. 2,299–315. With an appendix by Daniel S. Farley.[27] A. Joyal and R. Street, Braided tensor categories , Adv. Math. (1993), no. 1, 20–78.[28] , The geometry of tensor calculus. I , Adv. Math. (1991), no. 1, 55–112.[29] W. L¨uck, The type of the classifying space for a family of subgroups , J. Pure Appl. Algebra (2000), no. 2, 177–203.[30] S. Mac Lane, Categories for the working mathematician , 2nd ed., Graduate Texts in Math-ematics, vol. 5, Springer-Verlag, New York, 1998.[31] D. McDuff, On the classifying spaces of discrete monoids , Topology (1979), no. 4, 313–320.[32] N. Monod, Groups of piecewise projective homeomorphisms , Proc. Natl. Acad. Sci. USA (2013), no. 12, 4524–4527.[33] R. Par´e, Universal covering categories , Rend. Istit. Mat. Univ. Trieste (1993), no. 1-2,391–411 (1994).[34] D. Quillen, Higher algebraic K -theory. I , Springer, Berlin, 1973, pp. 85–147. Lecture Notesin Math., Vol. 341.[35] , Homotopy properties of the poset of nontrivial p -subgroups of a group , Adv. in Math. (1978), no. 2, 101–128.[36] P. Selinger, A survey of graphical languages for monoidal categories , New structures forphysics, Lecture Notes in Phys., vol. 813, Springer, Heidelberg, 2011, pp. 289–355.[37] E. H. Spanier, Algebraic topology , McGraw-Hill Book Co., New York-Toronto, Ont.-London,1966.[38] C. C. Squier, The homological algebra of Artin groups , Math. Scand. (1994), no. 1, 5–43.[39] M. Stein, Groups of piecewise linear homeomorphisms , Trans. Amer. Math. Soc. (1992),no. 2, 477–514.[40] R. W. Thomason, Cat as a closed model category , Cahiers Topologie G´eom. Diff´erentielle (1980), no. 3, 305–324.[41] , Homotopy colimits in the category of small categories , Math. Proc. Cambridge Phi-los. Soc. (1979), no. 1, 91–109.[42] W. Thumann, Operad groups , PhD thesis, KIT Karlsruhe, 2015. urn:nbn:de:swb:90-454145 .[43] M. Weiss, What does the classifying space of a category classify? , Homology Homotopy Appl. (2005), no. 1, 185–195.(2005), no. 1, 185–195.