aa r X i v : . [ m a t h . C T ] F e b MOMENT CATEGORIES AND OPERADS
CLEMENS BERGER
To Bob, in gratitude and friendship
Abstract.
A moment category is endowed with a distinguished set of splitidempotents, called moments, which can be transported along morphisms.Equivalently, a moment category is a category with an active/inert factorisa-tion system fulfilling two simple axioms. These axioms imply that the momentsof a fixed object form a monoid, actually a left regular band.Each locally finite unital moment category defines a specific type of operadwhich records the combinatorics of partitioning moments into elementary ones.In this way the notions of symmetric, non-symmetric and n -operad correspondto unital moment structures on Γ, ∆ and Θ n respectively.There is an analog of Baez-Dolan’s plus construction taking a unital mo-ment category C to a unital hypermoment category C + . Under this construc-tion, C -operads get identified with C + -monoids, i.e. presheaves on C + satisfy-ing Segal-like conditions strictly. We show that the plus construction of Segal’scategory Γ fully embeds into the dendroidal category Ω of Moerdijk-Weiss. Contents
Introduction. 2Acknowledgements 31. Moment categories 31.1. Active/inert factorisation systems 41.2. Pushing forward moments 51.3. Centricity 112. Unital moment categories 142.1. Units 142.2. Operads 182.3. Monoids 193. Hypermoment categories 213.1. Dendrices and graphices 233.2. Plus construction 243.3. Segal cores and extensionality 273.4. Monadicity 29References 30
Date : January 31, 2021.2020
Mathematics Subject Classification.
Primary ; Secondary .
Key words and phrases.
Moment category; Operad; Segal condition; Dendrix; Plus construc-tion; Monadicity.
Introduction.
What is an operad ? Although the pioneering work of May [29] and Boardman-Vogt [10] is almost half a century old, the question is more intricate than it mightseem at first sight. A multitude of types of operads have appeared (symmetric,non-symmetric, cyclic, modular, coloured, ...) and are used in different areas ofmathematics and even outside. A common feature is the existence of a process ofsubstitution. The present text proposes an unconventional approach, not based onthe concept of substitution, and not intended to enlarge further the panorama ofoperadic structures, but rather to look at them from a different perspective.Starting point is the existence of an active/inert factorisation system with suit-able properties. The basic example is Segal’s category Γ, the dual of the category offinite sets and partial maps. Our terminology has been motivated by Lurie [26] whouses extensively the inert/active factorisation system of Γ op . Active morphisms inΓ can be viewed as partitions of the target, indexed by the elements of the source.Inert morphisms are simply inclusions. Segal’s motivation [33] to choose Γ comesfrom the existence of a canonical covariant functor γ ∆ : ∆ → Γ linking simplicialcombinatorics to Γ. There is an active/inert factorisation system for ∆ compati-ble with this functor: active morphisms are endpoint-preserving, inert morphismsdistance-preserving. By means of a wreath product [7] the active/inert factorisationsystem carries over to Joyal’s categories Θ n [23].In all three examples, inert morphisms have unique active retractions. This pro-duces split idempotent endomorphisms, called moments , in bijective correspondencewith inert subobjects. We call the whole structure a moment category . Moreover,in all three cases, there is a well-defined object with a single centric moment, called unit . Inert subobjects are called elementary if they have a unit as domain. Itturns out that the scheme according to which moments decompose into elementarymoments, defines a operad-like structure, and this is so for any unital moment cat-egory. ∆-operads are non-symmetric operads, Γ-operads are symmetric operads,and Θ n -operads are Batanin’s n -operads [3].There exists an (essentially unique) augmentation γ C : C → Γ for each unitalmoment category C . This functor can be considered as a notion of cardinality .It also suggests that active morphisms in C are generalised partitions, and inertmorphisms generalised inclusions. Taking the existence of such an augmentationas basic leads to the more flexible notion of hypermoment category . There aretwo interesting examples of hypermoment categories. The dendroidal category Ωof Moerdijk-Weiss [30] and the graphoidal category Γ l of Hackney-Robertson-Yau[19]. In both cases, the inert morphisms are inclusions, often referred to as outerface operators . The active morphisms are either degeneracy operators or inner faceoperators . The latter can be understood geometrically as insertion of dendrices(resp. graphices) into vertices of dendrices (resp. graphices). The augmentationtakes a dendrix (resp. graphix) to its vertex set. Ω-operads (resp. Γ l -operads) aretree-hyperoperads (resp. graph hyperoperads), cf. Getzler-Kapranov [18].For brevity, algebras over the terminal C -operad are called C -monoids . The C -monoids are interesting in their own because they are “special” presheaves on theactive subcategory of C and are often algebraically simpler than operads. Thanksto the presence of inert morphisms, the notion of C -monoid can be reformulatedby means of strict Segal conditions. It is then natural to define C ∞ -monoids as OMENT CATEGORIES AND OPERADS 3 simplicial presheaves on C subject to homotopical Segal conditions. The tablebelow (copied from 2.13) summarises the notions we get in this way. C C -operad C -monoid C ∞ -monoid group-like C ∞ -monoidΓ sym. operad comm. monoid E ∞ -space infinite loop space∆ non-sym. operad assoc. monoid A ∞ -space loop spaceΘ n n -operad n -monoid E n -space n -fold loop spaceΩ tree-hyperoperad sym. operad ∞ -operad (stable ∞ -operad)Γ l graph-hyperoperad properad ∞ -properad (stable ∞ -properad)Most notably, symmetric operads appear twice, as Γ-operads and as Ω-monoids.This reveals a tight relationship between Γ and Ω which occurs implicitly at sev-eral places in literature [22, 3, 15]. We deduce this relationship from an analog ofthe plus construction of Baez-Dolan [1]. For hypermoment categories C , the plusconstruction C + is defined as a category of abstract C -trees, which are composablechains of active morphisms of C starting with a unit of C . The inert subcategoryof C contributes to the inert subcategory of C + . If C = Γ we get a full subcategoryΓ + of Ω. While Γ has a single unit, Ω and Γ + have a unit for each natural num-ber, namely the corolla with n leaves. It is a pleasant feature that the units of ahypermoment category are always determined intrinsically by the active/inert fac-torisation system. The plus construction converts basic operations of C into unitsof C + . Under the plus construction, C -operads get identified with C + -monoids.Therefore, homotopy C -operads can be modeled as ( C + ) ∞ -monoids.We show among others that if the hypermoment category C is extensional (whichis the case for all five categories of the table above), then C -trees can be insertedinto vertices of C -trees. This allows us to define a Feynman category F C [24] whosebasic morphisms are C -trees. The category of F C -algebras is then equivalent to thecategory of C -operads and this implies that the forgetful functor from C -operadsto C -collections is monadic.The plan of this text is as follows. In Section 1 we define moment categories andestablish their basic properties. The notion of centric moment is studied in somedetail because it enters in the definition of unit, and relates moment categoriesto restriction categories of Cockett-Lack [16]. Section 2 deals with unital momentcategories and defines their operads and monoids. Section 3 studies hypermomentcategories with emphasis on the dendroidal and graphoidal categories. We introducethe plus construction, and relate hypermoment categories to operadic categories ofBatanin-Markl [11] and to Feynman categories of Kaufmann-Ward [24].We have limited ourselves to the combinatorial aspects of moment categories andhope to pursue homotopical applications elsewhere. In Barwick’s article [3] muchabstract homotopy theory is developed in the setting of his operator categories.The recent preprint [14] of Chu-Haugseng is even closer to the present text. Acknowledgements.
This text would not exist without the many instructive andenlightening discussions I had over the years with Clark Barwick, Michael Batanin,Richard Garner, Ralph Kaufmann, Steve Lack, Ieke Moerdijk and Mark Weber.1.
Moment categories
This introductory section contains slightly more than absolutely necessary forthe application to operads in the following sections. From a purely abstract point of
CLEMENS BERGER view, moment categories can be considered as categorification of left regular bands,well known in semigroup literature, cf. [28]. It is remarkable that the constructionof the universal commutative quotient of a left regular band extends to momentcategories. Because the underlying notion of commutativity is quite subtle weintroduce the term centric for it.1.1.
Active/inert factorisation systems.
A class of morphisms in a category C is said to be closed if it is closed under composition and contains all isomorphismsof C . A subcategory of C is said to be wide if it contains all objects and allisomorphisms of C . Any closed class of morphisms in C defines a wide subcategoryof C , and conversely. For ease of exposition we will tacitly identify closed classeswith the corresponding wide subcategories.A ( C act , C in )- factorisation system on a category C consists of two closed classes C act and C in such that every morphism in C factors in an essentially unique wayas f = f in f act where f act belongs to C act and f in belongs to C in . Throughout thistext, the morphisms in C act will be called active , the morphisms in C in inert , andthe factorisation system itself will be called an active/inert factorisation system .The wide subcategory associated to C act (resp. C in ) is the active (resp. inert ) sub-category of C . The axioms of a moment category will imply that inert morphismsare split monomorphisms, that epimorphisms are active, but that in general activemorphisms do not need to be epimorphic. In Section 3 we will introduce hypermo-ment categories . These are equipped with an active/inert factorisation system withinert morphisms being monomorphisms but not necessarily split monomorphisms. Definition 1.1. A moment category is a category equipped with an active/inertfactorisation system such that (M1) every inert morphism has a unique active retraction; (M2) if f i = g for an inert morphism i and active morphisms f, g then f = gr where r is the unique active retraction of i provided by (M1) .A moment category is called centric if (MC) every active morphism has at most one inert section.A moment of an object A is an endomorphism φ of A such that if φ = φ in φ act then φ act φ in = 1 B for an object B . We shall say that the moment φ splits over B .A morphism is called retractive if it is active and admits an inert section.A moment functor is a functor preserving active and inert morphisms. It follows from (M1) and the essential uniqueness of active/inert factorisationsthat any morphism which is active and inert must be an isomorphism, hence theintersection C act ∩ C in is the closed class C iso of isomorphisms of C .We shall in general denote active morphisms by arrows of the form + ✲ andinert morphisms by arrows of the form > ✲ . Axiom (M2) is equivalent to thefollowing axiom (M2) ′ which is mnemotechnically easier to retain: OMENT CATEGORIES AND OPERADS 5 (M2) ′ If the left square below commutes then the right square as well
A f + ✲ B A f + ✲ BA ′ i ∧ ✻ + g ✲ B ′ i ′∧ ✻ A ′ r + ❄ + g ✲ B ′ + r ′ ❄ where r, r ′ are the unique active retractions of i, i ′ provided by (M1). Each moment of A satisfies φφ = φ in φ act φ in φ act = φ in φ act = φ and is thus a split idempotent endomorphism of A . We shall call an isomorphism class of inertmorphisms with fixed target A an inert subobject of A . Lemma 1.2.
For each object of a moment category there is a canonical bijectionbetween its moments and its inert subobjects.Proof.
Two splittings of an idempotent endomorphism take place over canonicallyisomorphic objects so that each moment of A defines an inert subobject of A .Conversely, by (M1), each inert morphism i : B → A generates a moment ir : A → A from which it derives, and isomorphic inert morphisms generate the samemoment. (cid:3) Pushing forward moments.
Moments have the advantage over inert sub-objects that there is no need to quotient by any equivalence relation. The momentsof an object form a subset of its endomorphism set.It is important to observe that while moments are in bijection with inert sub-objects it is in general not true that moments are also in bijection with retractivequotients, since a retractive morphism may have several distinct inert sections.Such a bijection holds if the moment category is centric , i.e. fulfills axiom (MC).It turns out that centric moment categories have already been studied in litera-ture since they correspond bijectively to split corestriction categories in the senseof Cockett-Lack [16]. We shall make this correspondence explicit, since it helps usto introduce useful terminology and notation, and gives us an alternative definitionof a moment category in terms of ... its moments !Let f : A → B be a morphism in a moment category and φ a moment of A .Choose a splitting φ = φ in φ act and denote the object over which φ splits by A ′ .Then factor the composite morphism f φ in : A ′ → B into an active morphism f ′ : A ′ → B ′ followed by an inert morphism ψ in : B ′ → B . We denote theunique active retraction of ψ in by ψ act : B → B ′ and the associated moment by ψ = ψ in ψ act .The pushforward of the moment φ along f is then defined by f ∗ ( φ ) = ψ . Thefollowing diagram summarises the construction: A f ✲ B with f ∗ ( φ in φ act ) = ψ in ψ act .A ′ φ act + ❄ φ in ∧ ✻ + f ′ ✲ B ′ ψ in ∧ ✻ + ψ act ❄ (1) CLEMENS BERGER
Observe that the isomorphism type of the morphism ψ in : B ′ → B (with fixed B )only depends on the isomorphism type of the morphism φ in : A ′ → A (with fixed A ), i.e. ψ is uniquely determined by the morphism f and the moment φ .By construction, the inner square is commutative. In order to show that theouter square is commutative as well, we decompose (1) as follows: A f act + ✲ B ′′ > f in ✲ BA ′ φ act + ❄ φ in ∧ ✻ + f ′ ✲ B ′ ξ in ∧ ✻ + ξ act ❄ ========== B ′ ψ in ∧ ✻ + ψ act ❄ (2)Observe that the active/inert factorisation of f φ in can be obtained by composingthe active/inert factorisation of f act φ in with f in , hence we can assume f in ξ in = ψ in .Therefore, the unique active retraction ξ act of ξ in is the composite of the uniqueactive retractions of f in and ξ in . In particular, ψ act f in = ξ act and the right handsquare is commutative. It suffices now to show that ξ act f act = f ′ φ act . This followsfrom axiom (M2) ′ since the left inner square of (2) commutes by construction.As corollary we obtain for each moment φ of A and each morphism f : A → B theimportant relation f φ = f ∗ ( φ ) f . Moreover, the essential uniqueness of active/inertfactorisations implies that pushforward is functorial in the following sense: for f : A → B and g : B → C we have ( gf ) ∗ ( φ ) = g ∗ ( f ∗ ( φ )). Finally, it follows fromthe definition of the pushforward that φ ∗ ( ψ ) = φψ for any two moments of thesame object. This leads to the following definition: Definition 1.3. A moment structure on a category consists in specifying for eachobject A , a set m A of special endomorphisms of A , called moments , and for eachmorphism f : A → B , a pushforward operation f ∗ : m A → m B such that thefollowing four axioms hold (for any A , any φ, ψ ∈ m A and f : A → B, g : B → C ) : (m1) 1 A ∈ m A (m2) φ ∗ ( ψ ) = φψ (m3) ( gf ) ∗ = g ∗ f ∗ (m4) f φ = f ∗ ( φ ) f A morphism f : A → B is called active (resp. inert ) if f ∗ (1 A ) = 1 B (resp. if f admits a retraction r : B → A such that f ∗ ( φ ) = f φr for all φ ∈ m A ).A moment of A is said to split over B if there exists i : B ⇆ A : r such that ir = φ and ri = 1 B . Lemma 1.4.
A category with moment structure enjoys the following properties: (i)
The moment set m A is a monoid under composition such that φψ = φψφ for all φ, ψ ∈ m A . In particular, each moment is idempotent . (ii) For any φ, ψ ∈ m A the relations ψφ = φ and φψ = ψ jointly imply φ = ψ . (iii) An endomorphism φ of A belongs to m A if and only if φ ∗ (1 A ) = φ .For any morphism f : A → B and any φ ∈ m A one has f ∗ ( φ ) = ( f φ ) ∗ (1 A ) . (iv) Epimorphisms are active. (v)
For any splitting i : B ⇆ A : r of a moment φ ∈ m A , the retraction r isactive and the section i is inert. In particular, r ∗ ( φ ) = 1 B and i ∗ (1 B ) = φ . (vi) The class of active (resp. inert) morphisms is closed. (vii) f has an active/inert factorisation if and only if the moment f ∗ (1 A ) splits. OMENT CATEGORIES AND OPERADS 7
Proof. –(i) By (m1) and (m2), the moment set m A is a monoid. (m2) and (m4) implythe relation φψφ = φψ . Putting ψ = 1 A yields φ = φ .(ii) By (i), φ = ψφ = ψφψ = φψ = ψ. (iii) By (m2), any moment satisfies φ = φ A = φ ∗ (1 A ). It follows then from(m3) that f ∗ ( φ ) = f ∗ ( φ ∗ (1 A )) = ( f φ ) ∗ (1 A ).(iv) By (m4), one has 1 B f = f A = f ∗ (1 A ) f . Thus if f is epic then f ∗ (1 A ) = 1 B .(v) Since r is active by (iv), we get by (iii) r ∗ ( φ ) = ( rφ ) ∗ (1 A ) = ( rir ) ∗ (1 A ) = r ∗ (1 A ) = 1 B . For each ψ ∈ m B , (m4) implies i ∗ ( ψ ) ir = iψr . Since m A is a monoid by (i), and ir = φ it follows that iψr ∈ m A . Therefore, by (iii) and (m2), iψr = ( iψr ) ∗ (1 A ) = i ∗ ( ψ ) so that i is inert. In particular, i ∗ (1 B ) = ir = φ .(vi) Both classes are closed under composition. Isomorphisms are active by (iv)and inert by (m4) since their pushforward action is the conjugation action.(vii) Assume first that f = ig with g active and i inert and denote by r aretraction of i such that ir is a moment of A . Then we get by (m3) and the definitionof active (resp. inert) morphisms that f ∗ (1 A ) = i ∗ ( g ∗ (1 A )) = ir . Conversely, if themoment f ∗ (1 A ) splits as f ∗ (1 A ) = ir , put g = rf . Since by (m4) f = f A = f ∗ (1 A ) f we get f = ig where i is inert by (v). Moreover, by (m3) and (v), we get g ∗ (1 A ) = r ∗ ( f ∗ (1 A )) = r ∗ ( ir ) = 1 hence g is active. (cid:3) Proposition 1.5.
A moment category is the same as a category with momentstructure in which all moments split.Proof.
We have seen that each moment category induces a moment structure byspecifying as moments those endomorphisms for which the active part is a retractionof the inert part. Indeed, the factorisation system defines a pushforward operationby diagram (1) above, which satisfies the axioms (m1), (m2), (m3), (m4). Notethat active (resp. inert) morphisms of the factorisation system are indeed active(resp. inert) in the sense of Definition 1.3, and that by definition all moments split.Conversely, given a category with moment structure in which all moments split,the active/inert factorisation system derives from Lemma 1.4(vi) and (vii). Theactive/inert factorisation is essentially unique because inert morphisms have re-tractions.It remains to be shown that the axioms (M1) and (M2) of a moment categoryhold. Assume that an inert morphism i : B → A has active retractions r, s : A → B with moments φ = ir and ψ = is . These moments of A are mutually “right-absorbing”, i.e. φψ = ψ and ψφ = φ . Therefore, by Lemma 1.4(ii), φ = ψ and hence r = s . Assume finally that f i = g for an inert morphism i and active morphisms f, g . Then for the (unique) active retraction r of i we get gr = f ir = f ∗ ( ir ) f = f where the last equality follows from the hypothesis that g and hence gr = f ir areactive so that by Lemma 1.4(iv), f ∗ ( ir ) = ( f ir ) ∗ (1) = 1. (cid:3) Definition 1.6.
Let φ, ψ be moments of the same object. The moment φ is said tobe a submoment of ψ if ψφ = φ in which case we write φ ≤ ψ .Two moments φ, ψ are said to be congruent if φ = φψφ and ψ = ψφψ , in whichcase we write φ ≃ ψ .A moment is called centric if its congruence class is singleton. A moment struc-ture is called centric if all its moments are centric. CLEMENS BERGER
The relation ≤ on m A is reflexive and transitive. By Lemma 1.4(ii) it is alsoantisymmetric and defines thus a partial order relation on the local monoid m A . Lemma 1.7.
Let A be an object of a category with moment structure. (i) If all moments of A are split, the poset ( m A , ≤ ) is isomorphic to the posetof inert subobjects of A ordered by inclusion; (ii) For any f : A → B and φ, ψ ∈ m A we have f ∗ ( φψ ) = f ∗ ( φ ) f ∗ ( ψ ) . Inparticular, pushforward f ∗ : m A → m B is order-preserving; (iii) Moments φ, ψ of A are congruent if and only if their active parts φ act , ψ act are isomorphic under A .Proof. (i) Consider split moments φ = ir and ψ = js . If i = jj ′ then φ = ir = jj ′ r and hence ψφ = φ . Conversely, if ψφ = φ then by (m2) and (m3) j ∗ s ∗ ( φ ) = φ and hence, by Lemma 1.4(v), js ∗ ( φ ) s = φ . Splitting the moment s ∗ ( φ ) as j ′ s ′ weget jj ′ s ′ s = ir and, since the last identity represents two splittings of the samemoment, we can assume without loss of generality that r = s ′ s and i = jj ′ .(ii) The second assertion follows from the first and the definition of the partialorders. For the first observe that f ∗ ( φψ ) 1 . ( iii ) = ( f φ ) ∗ ( ψ ) ( m = ( f ∗ ( φ ) f ) ∗ ( ψ ) ( m = f ∗ ( φ ) ∗ ( f ∗ ( ψ )) ( m = f ∗ ( φ ) f ∗ ( ψ ) . (iii) If φ = φψφ = φψ and ψ = ψφψ = ψφ then φ act ψ in ψ act = φ act and ψ act φ in φ act = ψ act . This implies that φ act ψ in and ψ act φ in induce inverse isomor-phisms between ψ act and φ act .Conversely, if ρψ act = φ act then φψ = φ in φ act ψ in ψ act = φ in ρψ act ψ in ψ act = φ ;dually, if σφ act = ψ act then ψφ = ψ . (cid:3) Remark . According to Lemma 1.4i the moments of any object of a moment cat-egory form a submonoid of the endomorphism monoid and fulfill the
Sch¨utzenbergerrelation φψφ = φψ . Such monoids are known in the semigroup literature as leftregular bands , cf. [28]. More precisely, a band is a semigroup consisting of idempo-tent elements, and a band is said to be left regular if the Sch¨utzenberger relationholds. In contrast to the semigroup literature, we shall always assume that a leftregular band has a neutral element, i.e. is a monoid. In the presence of a neutralelement, the idempotency of the elements follows from the Sch¨utzenberger relation.A morphism of left regular bands is required to preserve the multiplicative struc-ture, but not necessarily the neutral element. This is essential because pushforwardoperations is a morphism of left regular bands by Lemma 1.7ii, but not in general amorphism of monoids. Pushforward f ∗ : m A → m B preserves the neutral elementif and only if f : A → B is active , cf. Definition 1.3 and Proposition 1.5.In a category with moment structure each object has thus a local monoid m A ofmoments which is a left regular band. The partial order relation on m A (definedin Definition 1.6) is known in literature as Green’s L -relation. The congruencerelation is known as Green’s D -relation, and it is well-known that for any left regularband the quotient by the D -relation defines its universal commutative quotient. Inparticular, the local monoid m A is commutative if and only if its congruence relationis discrete, i.e. all moments of A are centric, cf. Definition 1.6.In general the quotient m A / ≃ is thus a commutative band. Commutative bandsare also known as meet-semilattices if multiplication is viewed as meet operation.The congruence class of 1 A serves as top element for the partial order. If the local OMENT CATEGORIES AND OPERADS 9 monoid m A is finite (which will always be the case for us) we get a finite meet-semilattice m A / ≃ with top element 1, and again it is well-known that one candefine a join operation x ∨ y by taking the meet of all z such that x ≤ z and y ≤ z .We get in this way a lattice with bottom element 0. Since the congruence class of1 A is singleton, we get a lattice with 0 = 1 as soon as A has non-identity moments.We are grateful to Steve Lack for having pointed out to us the following resultwhich is an important tool for constructing moment categories. Proposition 1.9.
Every category C with moment structure admits an idempotentcompletion into a moment category C whose objects are the moments of C . Con-gruent moments in C give rise to isomorphic objects in C .Proof. By definition, the objects of C are the moments of C , and for any pair ofmoments ( φ, ψ ) ∈ m A × m B , the morphism set is given by C ( φ, ψ ) = { f ∈ C ( A, B ) | ψf φ = f } . The identity of C ( φ, φ ) is φ , where composition in C is defined like in C . Weobviously get a category in this way.We now apply Proposition 1.5 in order to show that C is a moment category.The moment set of an object φ ∈ m A in C is given by m φ = C ( φ, φ ) ∩ m A = { ψ ∈ m A | φψφ = ψ } = { ψ ∈ m A | ψ ≤ φ } where the last identification follows from Lemma 1.4i.The pushforward operation f ∗ : m φ → m ψ for any f : φ → ψ in C is defined byrestricting the pushforward operation f ∗ : m A → m B of C because it follows from ψf φ = f that f ∗ takes any moment of A to a moment in m ψ . These restrictedpushforward operations fulfill the axioms (m1)-(m4) of a moment structure on C .It remains to be shown that the moments of C are split. Given ψ ∈ m φ , themoment ψ splits over the object φψ in C . Indeed, ψ : φ → φψ is a retraction withsection ψ : φψ → φ in C whose associated moment is ψ : φ → φ .For the second assertion observe that ψφ : φ → ψ and φψ : ψ → φ are mutuallyinverse morphisms in C if and only if φ and ψ are congruent moments in C . (cid:3) Examples . The following examples of moment categories are all locally finitewith a countable or finite set of objects. These categories play important roles inalgebraic topology or algebraic combinatorics. It is somehow surprising that theyshare the feature of carrying a moment structure.(a) The category Γ of SegalSegal’s category Γ [33] is the category of finite sets n = { , . . . , n } (where 0denotes the empty set) with operators m → n given by ordered m -tuples of pairwisedisjoint subsets of n . Composition is defined by( k ( M , . . . , M k ) ✲ m ( N , . . . , N m ) ✲ n ) = ( k ( S j ∈ M N j , . . . , S j k ∈ M k N j k ) ✲ n ) . We now define the following active/inert factorisation system on Γ:an operator f = ( N , . . . , N m ) : m → n is active if n = m F i =1 N i .an operator f = ( N , . . . , N m ) : m → n is inert if each N i is singleton.In particular, inert morphisms correspond to injections m → n and active mor-phisms can be considered as partitions of the target indexed by the elements of the source. It is now straightforward to check that this defines a factorisation systemon Γ fulfilling the axioms (M1), (M2), (MC) of a centric moment category.It is well-known that the dual category Γ op may be identified with (a skeletonof) the category of finite based sets and base-point preserving maps. Dualising theactive/inert factorisation system on Γ induces an inert/active factorisation systemon Γ op which has extensively been used by Lurie [26] in his theory of ∞ -operads.We borrowed our terminology from him, adding just the extra-term moment.The dual Γ op can also be identified with (a skeleton of) the category of finitesets and partial maps between them. In this setting, the inclusion of the active partΓ op act into Γ op can be interpreted as the inclusion of the category of finite sets and(total) set-mappings into the category of finite sets and partial set mappings. Thisinterpretation of Γ op illustrates well (the dual of) Proposition 1.14.We will see that the moment category Γ plays an important universal role insofaras every locally finite, unital moment category C comes equipped with an essentiallyunique augmentation γ C : C → Γ . This augmentation takes an object of C to theset of its elementary moments and views the morphisms of C as “partial partitions”of their target, cf. Proposition 2.6.(b) The simplex category ∆The simplex category ∆ is the category of finite non-empty ordinals [ m ] = { , . . . , m } , m ≥ , and order-preserving maps. A simplicial operator f : [ m ] → [ n ]is active if it is endpoint-preserving , i.e. f (0) = 0 and f ( m ) = n , and inert if it is distance-preserving , i.e. f ( i + 1) = f ( i ) + 1 for i = 0 , . . . , m − n ] → [0] areactive but have several inert sections for n > inner (resp. outer ) face operators.Inner face operators can be viewed as interval-partitions while outer face operatorscan be viewed as interval-inclusions. The simplex category admits thus a triplefactorisation system (degeneracy operator followed by inner face operator followedby outer face operator).(c) The categories Θ n of JoyalThe categories Θ n of Joyal [23] have several equivalent definitions. For n = 1 werecover example (b) since Θ = ∆. For general n >
0, the objects of Θ n are n -leveltrees. The maps can be described by first taking an n -level tree T to an n -globularset T ∗ , and then applying the free strict n -category functor F n from n -globular setsto strict n -categories. This leads to the following definition (cf. [6])Θ n ( S, T ) = nCat( F n ( S ∗ ) , F n ( T ∗ )) . An alternative way of defining Θ n is as an iterated wreath product of ∆ (cf. [7]).We will see below that for any two unital moment categories C , D there is a well-defined wreath product C ≀ D which is again a unital moment category. Since ∆ is aunital moment category, this permits to define iterated wreath products of ∆, andit turns out that Θ n is an n -fold wreath product ∆ ≀ · · · ≀ ∆ yielding the momentstructure of Θ n for free. OMENT CATEGORIES AND OPERADS 11
The active (resp. inert) morphisms of Θ n ( S, T ) have a geometric interpretationin terms of the n -level tree structures of S and T . The active morphisms correspondto tree-partitions of T labelled by S . The inert morphisms correspond to immersionsof S as a plain subtree of T . The inert morphisms are those belonging to the imageof the globular maps S ∗ → T ∗ under the free functor F n . The existence of thisactive/inert factorisation system on Θ n has been established in [6, Lemma 1.11].(d) Idempotent completion of a left regular bandWe refer the reader to Margolis-Saliola-Steinberg [28, Section 2] for more detailsconcerning the theory of left regular bands. Each left regular band L (cf. Section1.8) can be considered as a one-object category C L with endomorphism monoid thegiven left regular band L . If we define the unique moment set also to be equal to L ,and the pushforward operation to be left translation, then all axioms of a momentstructure are satisfied, axiom (m4) being the Sch¨utzenberger relation. Proposition1.9 applies and C L fully embeds into a moment category C L , the objects of which arethe elements of L . Moreover, congruent elements x, y in L give rise to isomorphic objects in C L . Therefore, C L is equivalent to a moment category b C L the objects ofwhich are the elements of the quotient-lattice L/ ≃ .The most prominent example of a left regular band is the face monoid L A of a hyperplane arrangement A in Euclidean space, cf. [28]. In this case, the quotient-lattice L A / ≃ coincides with the so-called intersection lattice I A of the hyperplanearrangement. The elements of the face monoid (“the facets”) can be identified withnon-empty intersections of the half-spaces delimited by the hyperplanes of A . Theproduct z = xy in L A of two facets is the first facet z crossed by a segment joiningan interior point of x to an interior point of y .The following proposition was suggested to us by Richard Garner. It shows thatthe category LRB of left regular bands (with neutral element) and morphisms ofleft regular bands is an example of a large moment category. Proposition 1.11.
The category
LRB is a moment category with active (resp.inert) morphisms the monoid morphisms (resp. the order-ideal inclusions).Each moment category C is endowed with a moment functor m C : C → LRB taking objects to their local moment monoids, and morphisms to the induced push-forward operation.Proof.
Each morphism of left regular bands f : M → N factors as a monoidmorphism f act : M → f (1) N followed by an order-ideal inclusion f in : f (1) N → N .This factorisation is essentially unique, because any order-ideal inclusion xN → N admits a monoid retraction N → xN given by left translation by x ; observe thatthe Sch¨utzenberger relation implies x ( yz ) = ( xy )( xz ), i.e. left translation preservesthe multiplicative structure. The uniqueness of this retraction yields axiom (M1)of a moment category. For (M2) observe that if a monoid morphism f : M → N remains a monoid morphism when restricted to the order-ideal i : xM → M then f ( x ) = 1, therefore f ir = f where r : M → xM is left translation by x .The second assertion follows from Lemmas 1.4i and 1.7ii and (m1), (m3). (cid:3) Centricity.
We now discuss in more detail centricity axiom (MC) and itsrelationship with corestriction structures of Cockett-Lack [16].
Lemma 1.12.
A moment is centric if and only if its inert part is the only inertsection of its active part.
Proof.
For inert sections i, i ′ of an active morphism r , the moments ir and i ′ r arecongruent. Therefore, if φ = φ act φ in is centric then φ in is the only inert section of φ act . Conversely, if the latter holds then for any congruence φ ≃ ψ , Lemma 1.7iiiyields ρ such that ρφ act = ψ act so that φ in = ψ in ρ whence φ = ψ in ρφ act = ψ . (cid:3) Proposition 1.13.
For a moment category the following conditions are equivalent: (i) the moment structure is centric; (ii) moments of the same object commute; (iii) centricity axiom (MC) holds.Proof.
We have seen in Section 1.8 that (i) implies (ii). Assume now (ii) andconsider an active morphism f : A → B with inert sections i, j : B → A . Then jf if = if jf implies jf = if and hence i = j , whence (iii). By Lemma 1.12 (iii)implies (i). (cid:3) Proposition 1.14.
Corestriction structures in the sense of Cockett-Lack [16] cor-respond one-to-one to centric moment structures . In particular, split corestrictioncategories are the same as centric moment categories.Proof.
A corestriction structure is the dual of a restriction structure (cf. [16, 2.1.1]).It is defined in terms of so-called cocombinators C ( A, B ) → C ( B, B ) : f f ∗ (1 A ) . These cocombinators extend to pushforward operations f ∗ : m A → m B by the rule f ∗ ( φ ) = ( f φ ) ∗ (1 A ) where m A = { φ ∈ C ( A, A ) | φ ∗ (1 A ) = φ } . Calling the elementsof m A moments, the axioms of Cockett-Lack can be stated as follows:(C1) f ∗ (1) f = f for any morphism f ;(C2) φψ = ψφ for any moments φ, ψ of the same object;(C3) ( ψf ) ∗ (1) = ψ f ∗ (1) for any morphism f and moment ψ of the target of f ;(C4) g f ∗ (1) = ( gf ) ∗ (1) g for any composable morphisms f and g .In [16, Lemma 2.1(iii)] Cockett and Lack deduce from (C1),(C2),(C3),(C4) that g ∗ ( φ ) = ( gφ ) ∗ (1) for any moment φ of the source of g . This implies our axiom(m3). Axiom (C4) then yields gφ = g ∗ ( φ ) g which is (m4). It follows from (C3)that the composite of two moments is a moment, which is equivalent to (m2), assoon as (m3) holds. Finally, (C1) implies (m1). By (C2) moments of the sameobject commute. Conversely, in a centric moment structure axiom (C2) holds,(C4) follows from (m4), (C1) follows from (C4) and (m1), while (C3) follows from(m2), (m3) and Lemma 1.4(iii). A corestriction category is split [16, 2.3.3] if andonly if all moments split. Therefore, Propositions 1.5 and 1.13 establish the secondstatement. (cid:3) Remark . This close relationship between corestriction and moment structureswas a surprise for us. However, in most examples relevant to operads, the mo-ment categories are not centric, but the splittings are needed for the definitionof the operad-type associated to a unital moment category. Therefore, startingfrom centric moment categories (i.e. split corectriction categories) our main con-cern consisted in dropping centricity (while keeping the splittings), whereas themain concern of Cockett-Lack was to drop the existence of splittings (while keepingcentricity). There are nevertheless similarities between both approaches.We have seen that every category with moment structure admits an idempotentcompletion turning it into a moment category. The existence of this idempotentcompletion relies (just as in the case of corestriction categories) on the completelyequational character of the axioms (m1)-(m4) of a moment structure.
OMENT CATEGORIES AND OPERADS 13
Our terminology differs from that of Cockett-Lack, partly in order to avoid theuse of too many co-s. The dictionnary is as follows:moment=corestriction idempotent, active=cototal, retractive=split corestriction.We end this introductory section by showing that each moment category has auniversal centric quotient.
Lemma 1.16.
For any moment category, the pushout of a retractive morphismalong an active morphism exists in the active subcategory and is again retractive.Proof.
Consider the following diagram
A f + ✲ B ❆❆❆❆❆❆❆ g ❯ A ′ r + ❄ i ∧ ✻ f ′ + ✲ B ′ r ′ + ❄ i ′∧ ✻❍❍❍❍❍❍❍ g ′ ❥ ....... h ❘ C in which i is any inert section of the retractive morphism r , and i ′ r ′ = f ∗ ( ir ). Weclaim that the retractive morphism r ′ : B + ✲ B ′ is the pushout in C act of r alongthe active morphism f : A + ✲ B . For this, we check the universal property andchoose two active morphisms g, g ′ such that g ′ r = gf . If h : B ′ + ✲ C with hr ′ = g exists then necessarily h = gi ′ . So there is no choice for h , and it remains to showthat hf ′ = g ′ , that h is active, and that hr ′ = g .Indeed, hf ′ = gi ′ f ′ = gf i = g ′ ri = g ′ . Since g ′ and f ′ are active and hf ′ = g ′ ,it follows from general properties of factorisation systems that h is active as well.Therefore, g and gi ′ = h are active, so that by axiom (M2), g = hr ′ as required. (cid:3) For a moment category C we define B C to be the category with same objects as C but with morphism-set ( B C )( A, B ) the set of isomorphism classes of cospans( f, r ) = ( A f + ✲ B ′ ✛ r + B )where f is active and r is retractive. Thanks to Lemma 1.16, these isomorphismclasses compose in the following way: ( g, s )( f, r ) = ( gf ′ , s ′ r ) where f ′ (resp. s ′ ) isthe pushout of f (resp. s ) along s (resp. f ) in the active subcategory of C .We thus get an identity on objects functor C → B C taking the morphism f : A → B to the cospan ( f act , r f ) where r f is the unique active retraction of f in .Two parallel morphisms f, g : A ⇒ B are said to be congruent (denoted f ≃ g )if they are identified under C → B C . By Lemma 1.7iii, this is the case if and onlyif the moments f ∗ (1 A ) and g ∗ (1 A ) are congruent in m B .Observe that B C comes equipped with an active/inert factorisation system forwhich isomorphism classes of cospans of the form ( f act ,
1) are active, and isomor-phism classes of cospans of the form (1 , r f ) are inert. It is straightforward to verifythat B C satisfies the axioms (M1), (M2), (MC) of a centric moment category. Proposition 1.17.
A moment category C is centric if and only if the functor C → B C is invertible. In general, the functor C → B C is initial among momentfunctors out of C taking values in centric moment categories. Proof.
The first statement is a consequence of the fact that each moment φ of anobject A equals φ ∗ (1 A ). For the second statement, let F : C → D be a functor withcentric target D and consider the following commutative diagram of functors C F ✲ DC / ≃ ❄ ✲ D / ≃∼ = ❄ in which the right vertical functor is an isomorphism because D is centric. Wetherefore get the required factorisation F : C → C / ≃ → D which is unique becausethe functor C → C / ≃ is identity on objects and full. (cid:3) Remark . It follows from Lemma 1.7iii that the categorical congruence restrictsto Green’s D -relation on m A for each object A . For congruent parallel morphisms f, g : A ⇒ B , the pushforwards f ∗ ( φ ) and g ∗ ( φ ) are congruent in m B for any φ ∈ m A . The first part of Proposition 1.17 is a Representation Theorem for centricmoment categories. Indeed, any centric moment category C is isomorphic to B C ,and is thus entirely determined by its active subcategory.The dual representation theorem for split restriction categories (cf. Remark 1.15)has been obtained by Cockett-Lack using different methods, cf. [16, Theorem 3.4].2. Unital moment categories
The purpose of this section is to single out a class of moment categories, called unital , in which moments decompose in a canonical way into elementary moments.A moment category is unital if there is a sufficient supply of so-called unit objects .The elementary moments are then those which split over a unit object.Each active morphism f : A → B induces a partition of the identity momentof B into moments f ∗ ( α ) where α runs through the elementary moments of A .This will allow us to associate, with every unital moment category of finite type,a specific operad type incorporating these partitions as its substitutional structure .We shall call operads of type C simply C -operads.The unital moment category Γ enjoys a universal property. Each unital momentcategory of finite type comes equipped with an essentially unique augmentation γ C : C → Γ, and Γ-operads turn out to be precisely symmetric operads in the senseof Boardman-Vogt and May. The augmentation γ C induces an adjunction betweenthe categories of C -operads and of Γ-operads so that C -operads can be symmetrized .The wreath product of Γ -augmented categories as defined in [7] takes a pair( C , D ) of unital moment categories to a unital moment category C ≀ D . We givehere an intrinsic description of this wreath product not depending on a chosenaugmentation over Γ and valid without finite type hypothesis.It follows from [7] that the operator categories Θ n of Joyal [23] are iteratedwreath products of the simplex category ∆. The unital moment structure on ∆induces thus a unital moment structure on Θ n . It turns out that the resultingΘ n -operads are precisely the ( n − -terminal n -operads of Batanin [3, 4] and thatour symmetrization functor coincides with Batanin’s [4] in this special case.2.1. Units.
We shall say that an object U of a moment category is primitive pro-vided U has non-identity moments, but none of them is centric. According to OMENT CATEGORIES AND OPERADS 15
Section 1.8, it amounts to the same to require that the universal semilattice quo-tient m U / ≃ is the two-element lattice { , } . Definition 2.1.
An object U of a moment category is called a unit if (U1) the object U is primitive; (U2) each active morphism with target U has one and only one inert section.A moment category is called unital if for each object A there exists an essentiallyunique pair ( U, φ ) consisting of a unit U and an active map φ : U + ✲ A .A moment is called elementary if it splits over a unit.Two moments of the same object are called disjoint if they do not share commonelementary submoments.The cardinality of an object is the cardinality the set of its elementary moments.A nilobject is an object of cardinality .A unital moment category is of finite type if the underlying category is small andevery object has finite cardinality. Lemma 2.2. – (i) Any inert (resp. active) morphism between units is invertible; (ii)
Elementary moments of the same object are either equal or disjoint.Proof. (i) Both assertions are equivalent. Let i : U > ✲ U ′ be an inert morphismbetween units. Then by (U2) i is the unique inert section of the active retraction r : U ′ + ✲ U . By Lemma 1.12 this implies that the moment ir of U ′ is centric sothat by (U1) we get ir = 1 U ′ .(ii) It suffices to show that comparable elementary moments are equal. ByLemma 1.7i, moments are comparable if and only if the associated inert subobjectsare comparable. For elementary moments these subobjects are units so that (i)allows us to conclude. (cid:3) Remark . All examples of Section 1.10 are unital moment categories:(a) Segal’s category Γ has a single unit, the one-element set 1. The elementarymoments of n correspond to inert subsets 1 > ✲ n , i.e. to elements of the set n .The cardinality of n is thus n and the only nilobject of Γ is 0. For every objet n there is a unique active morphism 1 + ✲ n .(b) The simplex category ∆ has a single unit, the segment [1]. The elementarymoments of [ n ] correspond to inert subobjects [1] > ✲ [ n ], i.e. to subsegments.The cardinality of [ n ] is thus n and [0] is the only nilobject of ∆. For every object[ n ] there is a unique active morphism [1] + ✲ [ n ].(c) Joyal’s category Θ n has a single unit, the linear n -level tree U n of height n .The elementary moments of an n -level tree T correspond to the vertices of T ofheight n . The nilobjects of Θ n are thus the n -level trees of height < n . Again, each n -level tree T receives a unique active morphism U n + ✲ T , cf. Proposition 2.8.(d) The idempotent completion b C M of a left regular band M has a single unit,represented by the neutral element 1 ∈ M . Every morphism in b C M is either inertor retractive or a moment. All objects different from 1 are nilobjects.From now on we fix a unital moment category C . The following lemma is themain reason for having introduced the notion of elementary moment. The argument is roughly speaking dual to the one establishing that for any set-mapping f : X → Y and x ∈ X there is a unique y ∈ Y such that x ∈ f − ( y ). Lemma 2.4.
For any elementary moment ψ of B and active morphism f : A + ✲ B there is a unique elementary moment φ of A such that ψ ≤ f ∗ ( φ ) .Proof. By definition, the elementary moment ψ factors through ψ act : B + ✲ U where U is a unit. By (U2) the composite active morphism ψ act f : A + ✲ U hasan inert section, thus defining an elementary moment φ of A . By Lemma 1.16,the moment f ∗ ( φ ) is associated with the pushout along f of ψ act f in C act , whence ψ ≤ f ∗ ( φ ).Assume we are given two elementary moments φ, φ ′ of A such that ψ ≤ f ∗ ( φ )and ψ ≤ f ∗ ( φ ′ ). In virtue of Lemma 1.16, ψ is then an elementary moment of thepushout Q of f ∗ ( φ ) act and f ∗ ( φ ′ ) act in C act . By a diagram chase, Q can be identifiedwith the pushout along f of the pushout P of φ act and φ ′ act in C act . We thus getan active morphism P + ✲ Q the target of which has an elementary moment. Itfollows that the source has then an elementary moment as well so that φ and φ ′ are not disjoint. According to Lemma 2.2ii this implies φ = φ ′ . (cid:3) Proposition 2.5.
Pushforward along any morphism f : A → B takes disjointmoments of A to disjoint moments of B . If f is active, then f ∗ induces a partitionof the set of elementary moments of B , indexed by the elementary moments of A .Proof. For the first assertion, we can assume that f is either inert or active. Twomoments are disjoint precisely when their associated inert subobjects do not sharean inert subobject with unital domain U . If f is inert, the pushforward oper-ation f ∗ associates to an inert subobject φ in : A ′ > ✲ A the inert subobject φ in f : A ′ > ✲ A > ✲ B , and hence pushforward along inert morphisms preservesdisjointness. If f is active, Lemma 2.4 shows that distinct elementary moments of A are taken to disjoint moments of B . This suffices by Lemma 2.2ii.For the second assertion we associate with each elementary moment e α of A theset of elementary moments e β of B such that e β ≤ f ∗ ( e α ). According to Lemmas2.2 and 2.4 this defines a partition of the set of elementary moments of B . (cid:3) In [33] Segal constructed a cardinality-preserving functor γ ∆ : ∆ → Γ and basedhis infinite delooping machine on the existence of this functor. The following propo-sition shows that Segal’s functor is actually a cardinality- and unit-preserving mo-ment functor and up to isomorphism uniquely determined by this property.
Proposition 2.6.
For each unital moment category of finite type C there is anessentially unique cardinality- and unit- preserving moment functor γ C : C → Γ .Proof. Each object of C has only finitely many elementary moments. This deter-mines γ C on objects since the category Γ has precisely one object for each finitecardinal. In particular, γ C is unit-preserving since units have cardinality 1 by (U1).In order to define the functor γ C on morphisms, we fix its image for each inertmorphism U > ✲ A with unital domain in such a way that distinct inert morphismsin C have distinct images in Γ. This amounts to fixing a bijection between theelementary moments of A and the elementary moments of γ C ( A ), i.e. to a totalordering of the elementary moments of A . Each moment φ of A determines then asubset of γ C ( A ), namely the one which corresponds to the elementary submomentsof φ . By Lemma 2.5, pushforward along f : A → B takes distinct elementary OMENT CATEGORIES AND OPERADS 17 moments of A to disjoint moments of B . We therefore get a well-defined map γ C ( f ) : γ C ( A ) → γ C ( B ) in Γ, and this assignment is easily seen to be functorial.Lemma 2.4 shows that γ C takes active morphisms to active morphisms. Sincepushforward along inert morphisms faithfully preserves elementary moments, γ C takes inert morphisms to inert morphisms so that γ C is indeed a moment functor.Conversely, any functor of moment categories γ C : C → Γ takes moments of A to moments of γ C ( A ), and pushforward operation f ∗ : m A → m B to pushforwardoperation γ C ( f ) ∗ : m γ C ( A ) → m γ C ( B ) . Therefore, once a bijection between theelementary moments of A and γ C ( A ) is fixed, there is no choice in defining γ C .Different choices of bijections lead to canonically isomorphic augmentations. (cid:3) Definition 2.7.
Let C , D be unital moment categories. The wreath product C ≀ D is defined to be the category for which • objects are tuples ( A, B α ) given by an object A of C and a family ( B α ) α ∈ el A of objects of D indexed by the set el A of elementary moments of A ; • morphisms are tuples ( f, f α ′ α ) : ( A, B α ) → ( A ′ , B ′ α ′ ) given by a morphism f : A → A ′ and morphisms f α ′ α : B α → B α ′ whenever α ′ ≤ f ∗ ( α ) in m A ′ . Proposition 2.8.
For unital moment categories C , D the wreath product C ≀ D is aunital moment category. In particular, Joyal’s Θ n are unital moment categories.Proof. We first define an active/inert factorisation system in C ≀ D where a morphism( f, f α ′ α ) is active (resp. inert) if f is active (resp. inert) in C and all f α ′ α are active(resp. inert) in D . For a given ( f, f α ′ α ) : ( A, B α ) → ( A ′ , B ′ α ′ ) we factor f as A f act + ✲ ˜ A f in > ✲ A ′ and observe that for each f α ′ α : B α → B α ′ with α ′ ≤ f ∗ ( α ) there is a unique ˜ α ∈ el ˜ A such that ( f in ) ∗ (˜ α ) = α ′ . Indeed, by Proposition 2.5 ( f act ) ∗ induces a partitionof el ˜ A indexed by el A so that each α ′ ≤ ( f in ) ∗ ( f act ) ∗ ( α ) is the pushforward of auniquely determined elementary submoment ˜ α ≤ ( f act ) ∗ ( α ). Factor now f α ′ α as B α ( f α ′ α ) act + ✲ ˜ B ( f α ′ α ) in > ✲ B ′ α ′ and index the middle object by ˜ α ∈ el ˜ A . This defines an active/inert factorisationof ( f, f α ′ α ) which is essentially unique by the essential uniqueness of the active/inertfactorisation systems in C and D . Axioms (M1) and (M2) for the wreath product C ≀ D follow from their validity in C and D .By axiom (U1) units have a single elementary moment, the identity. Therefore,units of C ≀ D are of the form ( U, V ) where U (resp. V ) is a unit of C (resp. D ).Axiom (U2) for ( U, V ) is a consequence of its validity in C (for U ) and in D (for V ).Finally, each object ( A, B α ) of C ≀ D receives an essentially unique active morphism( U, V ) + ✲ ( A, B α ).Joyal’s categories Θ n are iterated wreath products ∆ ≀· · ·≀ ∆ according to [7, The-orem 3.7]. The wreath product ∆ ≀ − of [7, Definition 3.1] coincides with Definition2.7 for C = ∆ using the unital moment structure of ∆, cf. Remark 2.3b. (cid:3) Remark . The augmentation γ Θ n : Θ n → Γ of Proposition 2.6 coincides up toisomorphism with the functor γ n : Θ n → Γ of [7, Definition 3.3].The inductive definition of γ n involves a so-called assembly functor α : Γ ≀ Γ → Γwhich takes ( n ; m , . . . , m n ) to m + · · · + m n , cf. [7, Lemma 3.2] and Remark 2.3a. This assembly functor induces wreath products of automorphisms in Γ entering intothe formulation of the equivariance properties of a symmetric operad.To be more precise, for ( σ ; τ , . . . , τ n ) ∈ Aut( n ) × Aut( m ) × · · · × Aut( m n ) wedenote the image under α : Γ ≀ Γ → Γ by σ ≀ τ i ∈ Aut( m + · · · + m n ). Explicitly,this internal wreath product σ ≀ τ i is obtained by postcomposing the direct sum τ ⊕ · · · ⊕ τ n with the obvious “block permutation” induced by σ .Thanks to Proposition 2.6, internal wreath products can be defined for any uni-tal moment category C . Let f : A + ✲ B be an active map in C and choosesplittings B α > ✲ B for the moments f ∗ ( α ) ∈ m B obtained by pushing forwardthe elementary moments α ∈ el A . For ( σ ; τ α ) ∈ Aut( A ) × Q α ∈ el A Aut( B α ) we shalldenote σ ≀ τ α any automorphism of B taken under γ C : C → Γ to the wreath product γ C ( σ ) ≀ γ C ( τ α ) in Γ. We shall not assume that these internal wreath products in C exist nor that they are unique if they exist, but this will often be the case.2.2. Operads.Definition 2.10.
Let C be a unital moment category of finite type and let E be asymmetric monoidal category with tensor product ⊗ and tensor unit I E .We choose for each elementary moment α ∈ el A , a splitting U α > ✲ A , i.e.an elementary inert monomorphism . The active/inert factorisation system inducesthen for each active morphism f : A + ✲ B inert morphisms B α > ✲ B splittingthe moments f ∗ ( α ) ∈ m B for α ∈ el A . These splittings are normalised in thefollowing sense: if f ∗ ( α ) = 1 B then B α = B and B α > ✲ B is the identity of B .A C -collection in E consists of a functor O : Iso( C ) → E . This yields foreach object A of C an Aut( A ) -object O ( A ) of E , and for each active morphism f : A + ✲ B in C , an object O ( f ) = N α ∈ el A O ( B α ) of E endowed with a canonical Q α ∈ el A Aut( B α ) -action.A C -operad in E is a C -collection in E equipped with structure maps • η U : I E → O ( U ) (one for each unit U of C ) • µ f : O ( A ) ⊗ O ( f ) → O ( B ) (one for each f : A + ✲ B in C act )such that the following unit, associativity and equivariance axioms hold: (i) for each f : U + ✲ A with unital domain, the composite morphism I E ⊗ O ( A ) η U ⊗ −→ O ( U ) ⊗ O ( f ) µ f −→ O ( A ) is a left unit constraint in E ;for each identity A + ✲ A , the composite morphism O ( A ) ⊗ O α ∈ el A I E ⊗ N η Uα −→ O ( A ) ⊗ O α ∈ el A O ( U α ) µ −→ O ( A ) is a right unit constraint in E ; (ii) for each pair A f + ✲ B g + ✲ C , the following diagram commutes: O ( A ) ⊗ O ( f ) ⊗ O ( g ) 1 ⊗ µ f,g ✲ O ( A ) ⊗ O ( gf ) O ( B ) ⊗ O ( g ) µ f ⊗ ❄ µ g ✲ O ( C ) µ gf ❄ OMENT CATEGORIES AND OPERADS 19 where µ f,g : O ( f ) ⊗ O ( g ) → O ( gf ) is obtained (after shuffling factors)as tensor product of the maps µ g α : O ( B α ) ⊗ O ( g α ) → O ( C α ) , α ∈ el A , induced by the following commutative diagram (cf. Proposition 2.5): A f + ✲ B g + ✲ CU α ∧ ✻ + ✲ B α ∧ ✻ + g α ✲ C α ∧ ✻ (iii) for each f : A + ✲ B and ( σ, τ α ) ∈ Aut( A ) × Q α ∈ el A Aut( B α ) admittinga wreath product σ ≀ τ α ∈ Aut( B ) , cf. Remark 2.9, the following diagram O ( A ) ⊗ O ( f ) µ f ✲ O ( B ) O ( A ) ⊗ O ( f ) O ( σ ) ⊗ N α O ( τ α ) ❄ µ fσ − ✲ O ( B ) O ( σ ≀ τ α ) ❄ commutes.The C -operads in E form a category Oper C ( E ) whose morphisms are maps of C -collections commuting with the structure maps η U and µ f . There is a restrictionfunctor γ ∗ C : Oper Γ ( E ) → Oper C ( E ) . Its left adjoint ( γ C ) ! : Oper C ( E ) → Oper Γ ( E ) (if existant) is called symmetrisation , cf. Remark 2.11 below.Remark . Let us review the examples of Section 1.10, cf. also Remark 2.3:(a) A Γ-collection O is a collection of objects O n endowed with Aut( n ) = Σ n -actions. These are often called symmetric collections or symmetric sequences. AΓ-operad is then precisely a symmetric operad in the sense of May [29] where themultiplicative structure µ f : O n ⊗ O m ⊗ · · · ⊗ O m n → O m + ··· + m n is associated tothe active map f : n + ✲ m + · · · + m n with partition ( m , . . . , m n ).Note that for C = Γ, our axioms (i), (ii) and (iii) correspond precisely to May’saxioms (b), (a) and (c) in [29, Definition 1.1], except that we drop O (0) = I E .(b) A ∆-operad is a non-symmetric operad . Our symmetrisation functor is theusual functor assigning a symmetric operad to a non-symmetric operad.(c) A Θ n -collection is a collection of objects indexed by n -level trees. Note thatthere are no non-trivial automorphisms in Θ n . It turns out that a Θ n -operad isprecisely a ( n − -terminal n -operad in the sense of Batanin [3, 4]. This followsfrom the observation that Batanin’s category of n -trees Ω n is the dual of Θ actn andthat for S + ✲ T in Θ actn , the induced partition of T is indexed by the fibres ofthe dual map in Ω n , cf. [4, Definition 4.3]. Our symmetrisation functor coincideswith Batanin’s [4, Section 8].2.3. Monoids.
For each C -operad O in E there is a natural notion of O -algebra .In this article, we will primarily be concerned with I C -algebras where I C is the unit C -operad defined by I C ( A ) = I E for each object A of C with structure maps inducedby the unit-constraints of E . Because of their importance I C -algebras will simplybe called C -monoids . We now make explicit what C -monoids are.We assume that E is a closed symmetric monoidal category with internal homhom E . Let us first define an endomorphism C -operad E X ∈ Oper C ( E ) for each functor X : C unit → E where C unit denotes the full subcategory of C act spannedby the units of C . According to Lemma 2.2ii, the subcategory C unit is a groupoid.For each object A of C we put E X ( A ) = hom E O α ∈ el A X ( U α ) , X ( U A ) ! where we have also chosen a fixed active morphism U A + ✲ A for each object A .By definition E X ( f ) = N α ∈ el A E X ( B α ) where U α > ✲ A f + ✲ B factors accordingto U α f α + ✲ B α > ✲ B . The multiplicative structure of the endomorphism operad µ f : E X ( A ) ⊗E X ( f ) → E X ( A ) ⊗ hom E O α ∈ el A ,β ∈ el Bα X ( U β ) , O α ∈ el A X ( U α ) → E X ( B )is then induced by tensor product and composition in E .For any C -operad O , an O -algebra in E is by definition a functor X : C unit → E equipped with a map of C -operads O → E X .Therefore, a C -monoid assigns to each unit U an object X ( U ) of E , and toeach object A a morphism N α ∈ el A X ( U α ) → X ( U A ) in E where U α is the chosensplitting of α ∈ el A and U A + ✲ A is the chosen active morphism.These data define a functor X : C op act → E such that X ( A ) = N α ∈ el A X ( U α ),and X ( f ) = N α ∈ el A X ( f α ) for f : A + ✲ B where f α : U α + ✲ B α is like above.In other words C -monoids are “special” presheaves X : C op act → E which in aprecise way are determined by their values at the units of C .In particular ∆-monoids (resp. Γ-monoids) in E are precisely associative (resp.commutative) monoids in E because ∆ op act (resp. Γ op act ) is the PRO (resp. PROP)for associative (resp. commutative) monoids, cf. Joyal [23] and Mac Lane [27].A Θ -monoid has two multiplicative structures (induced by the two 2-level treesof cardinality 2) sharing the same unit and distributing over each other. By theEckmann-Hilton argument Θ -monoids are equivalent to commutative monoids, asare Θ n -monoids for all n ≥
2. Below, Θ n -monoids will simply be called n -monoids .There is a weakening of the notion of C -monoid in E when E = Sets ∆ op is the category of simplicial sets . This simplicial weakening is based on a “good” notionof weak equivalence , including their stability under product and 2-out-of-3, as wellas the existence of a product preserving functor π : Sets ∆ op → Sets taking weakequivalences to bijections. Moreover, it is crucial that the category of simplicialsets is cartesian closed . Cartesian closedness implies that simplicial C -monoidsarise from simplicial presheaves C op → Sets ∆ op provided the latter satisfy strictSegal conditions . This leads to the following definition, see also Section 3.9 below. Definition 2.12. A C ∞ -monoid (for a unital moment/hypermoment category C )is a simplicial presheaf X : C op → Sets ∆ op such that (i) for nilobjects A , the value X ( A ) is terminal; (ii) for non-nilobjects A with representative set of elementary inert morphisms A α > ✲ A , the map s A : X ( A ) → Q α ∈ el A X ( A α ) is a weak equivalence. A simplicial presheaf on C satisfying just (i) will be called reduced . OMENT CATEGORIES AND OPERADS 21 If C = ∆ or C = Γ these structure maps s A are known as Segal maps sincethey were introduced in [33]. We shall use the same terminology for any unital(hyper)moment category C .A C ∞ -monoid yields a simplicial C -monoid provided all Segal maps are invertible.The set of path components of a C ∞ -monoid has the structure of C -monoid. For C = ∆ , Γ , Θ n a C ∞ -monoid X is called group-like if π ( X ) is a group. Examples . The following table illustrates the definitions of this section. To thespecial cases studied so far are added the dendroidal category
Ω of Moerdijk-Weiss[30] and the graphoidal category Γ l of Hackney-Robertson-Yau [19], cf. Section 3.1below. These are unital hyper moment categories to which all definitions of thissection apply (with a convenient adaptation of the group-like condition). C C -operad C -monoid C ∞ -monoid group-like C ∞ -monoidΓ sym. operad comm. monoid E ∞ -space infinite loop space∆ non-sym. operad assoc. monoid A ∞ -space loop spaceΘ n n -operad n -monoid E n -space n -fold loop spaceΩ tree-hyperoperad sym. operad ∞ -operad (stable ∞ -operad)Γ l graph-hyperoperad properad ∞ -properad (stable ∞ -properad)The last two columns should be interpreted as follows: there are two Quillenmodel structures on the category of reduced simplicial presheaves on C such that thefibrant objects are respectively C ∞ -monoids and grouplike C ∞ -monoids, and thehomotopy category is equivalent to the homotopy category of the claimed objects.For ∆ , Γ these results go back to Segal [33] and Bousfield-Friedlander [12], cf.also Boavida-Moerdijk [11]. For Θ n the E n -model structure may be obtained byrestricting Rezk’s model structure for weak n -categories [32] to reduced simplicialpresheaves, while the model structure for n -fold loop spaces is described in [7].The last two rows are more involved because C -monoids have a more complicatedstructure here. The Segal model structure for ∞ -operads has been described byCisinski-Moerdijk [13] (restricting it to reduced simplicial presheaves). Relatedmodel categories have been investigated by Barwick [3] and Chu-Haugseng-Heuts[15]. The model structure for ∞ -properads can be obtained by restricting the modelstructure of Hackney-Robertson-Yau [20].It would of course be desirable to have a uniform proof of existence for thesemodel structures. The key idea is to homotopically invert the Segal maps startingfrom a projective (or injective) model structure on the category of reduced simplicialpresheaves on C . We hope to come back to this topic in a future paper.3. Hypermoment categories
This section introduces hypermoment categories, which are more general thanmoment categories. A hypermoment category comes equipped with an active/inertfactorisation system where there is no correspondence between inert subobjects andmoments, but just an augmentation γ C : C → Γ compatible with the active/inertfactorisation systems. This induces a well-behaved notion of cardinality for theobjects of C and is enough to define C -operads and C -monoids like in Section 2.2.We then define a plus construction C + for unital hypermoment categories C withthe characteristic property that C -operads get identified with C + -monoids. Like in the original plus construction of Baez-Dolan [1] this is achieved by taking“basic operations” (i.e. active morphisms with unital domain) in C to “types” (i.e.units) in C + , and “reduction laws of operations” in C to “operations” in C + .It turns out that C + can be constructed as a category of special elements of thesimplicial nerve of C . A closely related construction for operator categories hasbeen considered by Barwick [3] and further studied by Chu-Haugseng-Heuts [15].Our plus construction C + is in general different from theirs because it also dependson the inert subcategory of C inexistant in an operator category.The dendroidal category of Moerdijk-Weiss [30] and the graphoidal categoryof Hackney-Robertson-Yau [19] are examples of hypermoment categories which arenot moment categories because both categories have inert morphisms without activeretraction. Nonetheless both are augmented over Γ by assigning to a dendrix (resp.graphix) its vertex set . We will show that Ω contains the plus construction Γ + ofSegal’s category Γ as a full subcategory. This is the reason for which in Table 2.13symmetric operads appear twice: as Γ-operads and as Ω-monoids.We finally deduce from the existence of the plus construction C + a generalmonadicity result for C -operads, viewed as structures on C -collections. As an in-termediate step we define for each unital hypermoment category C a Feynman cat-egory F C (cf. Kaufmann-Ward [24]) such that C + -monoids (and hence C -operads)get identified with F C -algebras. Definition 3.1. A hypermoment category is a category C equipped with an ac-tive/inert factorisation system and an augmentation γ C : C → Γ such that (i) γ C preserves active (resp. inert) morphisms; (ii) γ C preserves cardinality in the following sense: for each object A of C and each element > ✲ γ C ( A ) , there is an essentially unique inert lift U > ✲ A in C such that U satisfies unit-axiom (U2) of Definition 2.1.A hypermoment category is called unital if every object A of C receives an es-sentially unique active morphism U + ✲ A whose domain U is a unit of C , i.e.belongs to γ − C (1) and satisfies unit-axiom (U2) of Definition 2.1. According to Proposition 2.6, each unital moment category C admits an es-sentially unique augmentation γ C : C → Γ turning it into a unital hypermomentcategory. The essential difference between the two notions is that in a hypermo-ment category there might exist inert morphisms without active retraction. Thisimplies that for a given object A of a unital hypermoment category C there mightexist distinct units for the “global structure” U + ✲ A and the “local structures” U i > ✲ A . This possibility is excluded in a unital moment category because com-posing the “global structure” U + ✲ A with the active retractions r i : A + ✲ U i of the “local structures” yields active morphisms U + ✲ U i between units whichare necessarily invertible by Lemma 2.2i.Despite of this extra-freedom available in a unital hypermoment category, thedefinitions leading to the notion of C -operad, resp. C -monoid can be copied almostverbatim, provided we use instead of elementary moments elementary inert sub-objects . Indeed, the crucial pushforward operation of Section 1.2 applies to inertsubobjects without any modification. In particular, from now on, we shall denoteby el A the set of elementary inert subobjects of A , i.e. the set of isomorphismclasses of inert morphisms with unital domain and fixed codomain A .Before turning to examples we relate hypermoment and operadic categories. OMENT CATEGORIES AND OPERADS 23
Proposition 3.2.
For each unital hypermoment category ( C , C → Γ) of finite typein which units have no non-trivial automorphisms, the pair ( C op act , C op act → Γ op act ) carries the structure of an operadic category in the sense of Batanin-Markl [5] .Proof. Recall from Example 1.10a that Γ op act is a skeleton of the category of finitesets and set mappings. Moreover, each connected component of C op act containsan essentially unique unit. Such a unit is terminal in its connected componentof C op act because of the absence of non-trivial automorphisms. By the axiom ofchoice ( C being small) we can choose simultaneously a unit in each connectedcomponent. The abstract fibre structure of an operadic category is now obtainedas the dual of pushing forward elementary inert subobjects, thanks to Definition3.1ii. In particular, the transitivity of the pushforward operation yields axioms (iv)and (v) of an operadic category, cf. the ingredients of our Definition 2.10ii. (cid:3) Remark . Our notion of C -operad is almost the same as Batanin-Markl’s notionof operad over the operadic category C op act . The difference concerns equivariancewhich is missing in [5]. Nonetheless, an operad over an operadic category mayhave symmetries which are induced by the operad multiplication. Our equivarianceaxiom amounts to requiring that these “external” symmetries coming from theoperad multiplication coincide with the “internal” symmetries coming from theautomorphisms of C . So, the principal difference is that in our approach C -operadsare viewed as structures on collections with symmetries while in the approach ofBatanin-Markl they are viewed as structures on collections without symmetries.Batanin shows in [4, Proposition 3.1] that for symmetric operads both view pointsare equivalent.It would be interesting to characterise those operadic categories which comefrom unital hypermoment categories, and also to generalise the notion of operadiccategory in such a way that units with symmetries can be handled as well.3.1. Dendrices and graphices.
The dendroidal category
Ω has been introducedby Moerdijk-Weiss [30], see also [21] for a recent presentation.The objects of Ω (the dendrices ) are finite rooted trees, the morphisms of Ω aredefined by viewing such trees as coloured symmetric operads, where the coloursare the edges of the tree, and the operations are freely generated by the vertices ofthe tree. The morphisms of Ω are thus maps of coloured symmetric operads. Thedendrices may have vertices with a single incident edge (i.e. stumps ) representingconstant operations of the induced coloured symmetric operad.The simplex category ∆ may be identified with the full subcategory of Ω spannedby the linear trees without stumps. The moment structure of ∆ canonically extendsto a hypermoment structure of Ω where γ Ω : Ω → Γ takes a dendrix to its vertex-set.Mixing the active/inert factorisation system with the epi/mono factorisationsystem induces a triple factorisation system for Ω: each morphism can be writtenin an essentially unique way as a degeneracy operator followed by an inner faceoperator followed by an outer face operator.Degeneracy operators (i.e. retractive morphisms) correspond to dropping ver-tices with exactly two incident edges. Outer face operators (i.e. inert morphisms) S > ✲ T can be viewed as tree embeddings . Inner face operators (i.e. activemonomorphisms) S + ✲ T are dual to inner edge contractions of T and can beviewed as partitions of T into subtrees T α > ✲ T indexed by the vertex set of S .Alternatively, an inner face operator S + ✲ T can also be viewed as an insertion of trees T α into the vertices α of S such that T is the result of this insertion process.The active subcategory of Ω is generated by degeneracies and inner face operators.In contrast to ∆, the dendroidal category Ω has non-trivial symmetries, the treeautomorphisms. Also, in contrast to ∆, there are inert morphisms without activeretraction, namely those S > ✲ T for which the complement of S in T does not decompose into a coproduct of linear trees.The dendroidal category has a single nilobject: the edge | without vertices. The units of Ω are precisely the corollas C n where n is the number of leaves. Unit-axiom(U2) is satisfied because the source of a degeneracy S + ✲ C n has a unique vertexmapping to the vertex of C n so that there is a uniquely determined inert section C n > ✲ S . Moreover, this property characterises corollas. Axiom (ii) of Definition3.1 says then that for each vertex α of a dendrix T there is an essentially uniquepair consisting of a corolla C n ( α ) and an outer face operator C n ( α ) > ✲ T , where n ( α ) is the valency of α . The dendroidal category is unital because for each dendrix T there is also an essentially unique pair consisting of a corolla C n ( T ) and an innerface operator C n ( T ) + ✲ T where n ( T ) is the number of leaves of T .It can now be checked that Ω-operads in the sense of Definition 2.10 are preciselytree- hyperoperads in the spirit of Getzler-Kapranov [18]. This analogy motivatedour terminology of hypermoment category. It can also be checked by hand thatΩ-monoids are precisely (single-coloured) symmetric operads.Hackney, Robertson and Yau [19] further embed the dendroidal category Ω intoa graphoidal category Γ l . Its objects (the graphices ) are finite connected graphswith directed edges and directed leaves so that there are no oriented edge-cycles inthe graph. Each dendrix defines a graphix by directing all edges towards the root.The morphisms of Γ l are definable like the morphisms of Ω. It is best to describedirectly the triple factorisation system for Γ l , cf. [19]. Degeneracies correspondto dropping vertices with exactly one incoming and one outgoing edge. Outer faceoperators (i.e. inert morphisms) are graphix embeddings, while inner face operators(i.e. active monomorphisms) correspond to insertion of graphices into vertices ofgraphices. The augmentation Γ l → Γ takes a graphix to its vertex set.There is a single nilobject, the directed edge l without vertices. The unitsare directed corollas C m,n with m incoming leaves and n outgoing leaves. Thegraphoidal category is a unital hypermoment category because for each graphix G there are essentially unique inert morphisms C m ( α ) ,n ( α ) > ✲ G , resp. activemorphism C m ( G ) ,n ( G ) + ✲ G determined by the vertices α , resp. the leaves of G .We get full embeddings of hypermoment categories ∆ ֒ → Ω ֒ → Γ l .Γ l -operads are directed graph hyperoperads in the spirit of Getzler-Kapranov[18] while Γ l -monoids are (set-based) properads in the sense of Vallette [34] as canbe checked by hand. The reader should note that the morphism-set Γ l ( G, H ) iscontained in, but in general not equal to, the set of properad morphisms betweenthe set-based properads freely generated by G and H , cf. [19, Sections 5-6].3.2. Plus construction.
We present an analog of the plus construction C + ofBaez-Dolan [1] for unital hypermoment categories C . Its characteristic propertyis that C -operads get identified with C + -monoids. In particular, we obtain anequivalence of categories Γ + ≃ Ω red closely related to constructions of [22, 2, 15]. Definition 3.4.
Let C be a unital hypermoment category. OMENT CATEGORIES AND OPERADS 25 • A C -tree is a pair ([ m ] , A + ✲ · · · + ✲ A m ) consisting of an object [ m ] of ∆ and a functor A • : [ m ] → C act such that A is a unit of C ; • A C -tree morphism is pair ( φ, f ) consisting of a morphism φ : [ m ] → [ n ] in ∆ and a natural transformation f : A → Bφ which is pointwise inert,i.e. f i : A i → B φ ( i ) is inert in C for all i ∈ [ m ] ;A C -tree morphism ( φ, f ) is called active (resp. inert ) if φ is active and f isinvertible (resp. if φ is inert).A vertex of ([ m ] , A • ) is an elementary inert subobject U > ✲ A i for some i < m .Vertices will be represented by inert morphims ([1] , U + ✲ A ) > ✲ ([ m ] , A • ) where U + ✲ A > ✲ A i is the active/inert factorisation of U > ✲ A i + ✲ A i +1 .The plus construction C + is the category of C -trees and C -tree morphisms. Proposition 3.5.
The plus construction takes unital hypermoment categories tounital hypermoment categories. The augmentation γ C ∗ : C + → Γ takes a C -tree toits vertex set. Units of C + are C -trees ([1] , U + ✲ A ) where U is a unit of C .Proof. Every C -tree morphism ( φ, f ) : ([ m ] , A • ) → ([ n ] , B • ) factors as an activemorphism ( φ act ,
1) : ([ m ] , A • ) → ([ m ′ ] , A • φ act ) followed by an inert morphism( φ in , f φ act ) : ([ m ′ ] , A • φ act ) → ([ n ] , B • ). This factorisation is essentially uniquebecause it is unique on the first factor and essentially unique on the second factorthanks to the essential uniqueness of the active/inert factorisation system of C .The augmentation γ C + : C + → Γ is defined on objects by sending a C -tree toits vertex set. In order to extend this to a functor we first determine the units of C + . By definition, these are the objects of C + with a unique vertex and subjectto unit-axiom (U2). Since projection on the first factor C + → ∆ preserves activeand inert morphisms, and reflects retractive morphisms, any unit of C + must beof the form ([1] , A • ) with active map A + ✲ A and such that A is a unit of C .Such an object ([1] , A • ) satisfies unit-axiom (U2) because active retractions in C + are pointwise invertible. The vertex set of ([1] , A • ) is singleton.It follows that the vertex set of a C -tree ([ m ] , A • ) may be identified with the set ofelementary inert subobjects of ([ m ] , A • ). The proofs of Propositions 2.5 and 2.6 arenow applicable in our context and permit to extend γ C + : C + → Γ to morphisms.By construction, this augmentation satisfies axioms (i) and (ii) of Definition 3.1.For an arbitrary C -tree ([ m ] , A • ) the total composition A + ✲ A m yields aunit ([1] , A + ✲ A m ) of C + . By unitality of C , the C -tree ([ m ] , A ) receives anessentially unique active map from ([1] , A + ✲ A m ) in C + . (cid:3) Theorem 3.6.
For each unital hypermoment category C of finite type, the cate-gories of C -operads and of C + -monoids are equivalent.Proof. We first show that the data underlying a C -operad and a C + -monoid areequivalent. It follows from Section 2.3 and Proposition 3.5 that a C + -monoidis a functor X : ( C + act ) op → E such that for each C -tree we have X ([ m ] , A ) = N α ∈ γ C ∗ ([ m ] ,A • ) X ([1] , U α + ✲ A α ) where ([1] , U α + ✲ A α ) > ✲ ([ m ] , A • ) is avertex. Since C is unital, the functor which associates to a unit ([1] , U α + ✲ A α )of C + the object A α of C induces an equivalence of categories ( C + ) unit ∼ → Iso( C ) sothat the underlying objects of a C -operad and a C + -monoid are indeed equivalent.Let us denote O X : Iso( C ) → E the corresponding underlying object of a C -operad.For units U of C , O X ( U ) has a distinguished element because X ([1] , U ) has one.The multiplication µ f : O X ( A ) ⊗ O X ( f ) → O X ( B ) for an active map f : A + ✲ B in C corresponds to the X -action of the active morphism([1] , U + ✲ B ) + ✲ ([2] , U + ✲ A f + ✲ B )in C + act . Unit-, associativity- and equivariance axioms of an O -operad O X are thencodified by the X -action of certain commutative diagrams in C + act . For instance, theassociativity constraint is induced by the commutativity of the following diagram([3] , U + ✲ A f + ✲ B g + ✲ C ) ([2] , U + ✲ A gf + ✲ C ) o o ([2] , U + ✲ B g + ✲ C ) O O ([1] , U + ✲ C ) O O o o in C + act . A consistent X -action with respect to these commutative diagrams isenough to define a C + -monoid X : ( C + act ) op → E since simplicial nerves are de-termined by their 3-skeleton. Under this correspondence between C -operads and C + -monoids the respective morphisms correspond as well. (cid:3) For the next proposition we have to consider several variants of the dendroidalcategory Ω compatible with the active/inert factorisation system. A dendrix iscalled open if it has no stumps. The active/inert factorisation system restrictsto the full subcategory of open dendrices. This subcategory contains all corollaswith the exception of C . We call a dendrix reduced if it is either open or has theproperty that its stumps are precisely the vertices of highest level. In particular thenullary corolla C is reduced. It follows from the next proposition (but can also bechecked by hand) that the full subcategory Ω red spanned by the reduced dendricesis closed under the active/inert factorisation system. A dendrix is called planar iffor each vertex the set of incoming edges is equipped with a linear order. Proposition 3.7.
The plus construction Γ + (resp. ∆ + ) is equivalent to the hyper-moment category of reduced dendrices (resp. reduced planar dendrices).Proof. We shall show that Γ-trees are reduced dendrices, and that morphisms ofΓ-trees are the same as morphisms of dendrices. The planar case is similar.Let ([ m ] , A + ✲ · · · + ✲ A m ) be a Γ-tree. We consider the object n of Γ asan n -element-set and represent the active maps as partitions of the target (withpossibly empty parts) indexed by the elements of the source. Such a partition canbe described as a bipartite graph in an evident way (namely the graph of the dualset mapping). These bipartite graphs of a Γ-tree glue together along vertices of theΓ-tree and define in this way a dendrix. The resulting dendrix is necessarily reducedbecause the only way of having stumps is to have A i = A i +1 = · · · = A m = 0 with A i − = 0 in which case the Γ-tree has A i − stumps lying on the top level ofthe dendrix. Each reduced dendrix has an essentially unique such presentationas Γ-tree. One checks now that codimension one active/inert monomorphisms in C + act correspond to codimension one outer/inner face operators in Ω, and retractivemorphisms in C + act correspond to degeneracy operators in Ω. Moreover compositionis the same on both sides, and a morphism in C + act is uniqely determined by whatis does on edges like the morphisms in Ω. (cid:3) Remark . The subcategory ∆ of Barwick’s category ∆ F [2] considered by Chu-Heuts-Haugseng [15] is equivalent to our Γ + for the following three reasons: (a) OMENT CATEGORIES AND OPERADS 27 Γ op act is equivalent to F; (b) in Γ retractive morphisms have unique inert sections bycentricity so that the duals of these retractive morphisms compose like our inertmaps; (c) the pullback condition of [15] is our Lemma 1.16.In [15, Section 4], a functor ∆ → Ω is constructed which upon inspectioninduces the equivalence ∆ ≃ Ω red of Proposition 3.7. By [15, Theorem 5.1], theresulting inclusion Ω red ֒ → Ω (leaving out all non-reduced dendrices) induces anequivalence of Segal type homotopy theories for simplicial preheaves on both sides.This relates to work of Heuts-Hinich-Moerdijk [22] where still another category ofdendroidal forests is used to compare the Cisinski-Moerdijk model structure [13]for simplicial presheaves on Ω with Lurie’s model for ∞ -operads [26].The planar version of Proposition 3.7 is related to Baez-Dolan’s n -opetopes [1].Starting with the simplex category ∆ we can apply n times our plus constructionand recover n -opetopes as the units of the resulting hypermoment category. As ageneral fact, for any hypermoment category over Γ (resp. ∆), the plus constructioncomes equipped with a functor C + → Γ + (resp. C + → ∆ + ). In particular, C -trees have an underlying Γ-tree (resp. ∆-tree) which is a reduced dendrix (resp. areduced planar dendrix). This is one way to see why opetopes are “trees of trees oftress of ...”. An interesting feature of our approach to opetopes is the presence ofinert morphisms and degeneracies which might reveal hidden aspects of opetopes.3.3. Segal cores and extensionality.
We mention here two properties whichare present in all hypermoment categories so far discussed. The first permits areformulation of the Segal conditions, cf. Definition 2.12. The second will allow usto define insertion C -trees into vertices of C -trees.For the notion of dense subcategory, see e.g. [8], especially Lemma 1.7 therein. Definition 3.9.
The
Segal core of a hypermoment category C is the full subcategoryof the inert subcategory spanned by the units and the nilobjects.A unital hypermoment category is called strongly unital if its Segal core is dense in the inert subcategory. This means that each object of C , when viewed as an object of the inert sub-category C in , is a canonical colimit of unit- and nilobjects. A simplicial presheaf X : C op → Sets ∆ op is then said to satisfy the Segal conditions strictly (resp. weakly)if its restriction to the inert subcategory takes the density colimit cocones to limitcones (resp. homotopy limit cones) in simplicial sets. The advantage of this refinedSegal condition is that it applies to general simplicial presheaves on C . If the lat-ter are reduced (cf. Definition 2.12i) then the (homotopy) limit cones are actually(homotopy) product cones, and we recover the Segal condition of Definition 2.12ii.Let us indicate the Segal cores of our main examples: C Γ ∆ Θ n Ω Γ l Segal core 0 → ⇒ [1] cell-inclusions edge-inclusions edge-inclusionsof C of glob. n -cell of corollas of dir. corollasStrong unitality of Γ and ∆ have been used by Segal [33]. Strong unitality ofΘ n has been used by Batanin [3] and the author [6] to decompose an n -level treeinto a canonical colimit of its linear subtrees. These colimit cocones induce thecanonical decomposition of an n -dimensional globular pasting scheme into globularcells, cf. Leinster [25]. Strong unitality of Ω (resp. Γ l ) translates into a canonicaldecomposition of dendrices into vertex-corollas (resp. of graphices into directed vertex-corollas). In all five cases, these colimit decompositions enter into the Segalmodel structure for simplical presheaves on C , cf. [32, 13, 20]. Definition 3.10.
A hypermoment category C is called extensional if elementaryinert morphisms admit pushouts along active morphisms and these pushouts areinert and preserved by the augmentation. In Γ these pushouts exist (they are dual to pullbacks of partial identities). In∆ as well, and using the wreath product, in Θ n too. A direct inspection showsthat they exist in Ω and Γ l . In all these cases, inert morphisms are generated(under composition and pushout) by elementary inert morphisms so that we getthe existence of pushouts of general inert morphisms along active morphisms andthese are again inert. This property of an active/inert factorisation system is dualto what is known in literature as a modality . It is also worthwhile noting thatextensionality of ∆ is the building block of the theory of decomposition spaces ofG´alvez-Kock-Tonks [17] so that it is conceivable that a similar theory exists for anyextensional hypermoment category. Proposition 3.11.
The plus construction of a unital (resp. extensional) hypermo-ment category is strongly unital (resp. extensional).Proof.
We first determine the Segal core of C + . We have seen that the units of C + are of the form ([1] , U + ✲ A ) where A is any object of C . The nilobjects of C + are of the form ([0] , U ). Inert morphisms between such objects are induced byautomorphisms of C and inert morphisms of the form V > ✲ A . In the latter case,the units ([1] , U + ✲ A ) and ([1] , V + ✲ B ) of C + are said to be linked by theelementary cospan ([1] , U + ✲ A ) ✛ ✛ ([0] , V ) > ✲ ([1] , V + ✲ B ) in ( C + ) in .Now, given any C -tree ([ m ] , A + ✲ · · · + ✲ A m ), its vertex set is the set ofinert morphisms of the form ([1] , U + ✲ A ) > ✲ ([ m ] , A + ✲ · · · + ✲ A m ). Wesay that two vertices are linked if there exists an elementary cospan between theirdomains inducing a commuting square. The image of a C -tree under γ C : C → Γ isa Γ-tree, i.e. a dendrix. Under this mapping, vertices are sent to vertices, and twovertices of a C -tree are linked if and only if their image-vertices are so, thanks toDefinition 3.1ii of a hypermoment category. For a dendrix, two vertices are linkedif and only if there exists a well-defined edge joining the two vertices.This implies that the vertex/edge -structure of ([ m ] , A + ✲ · · · + ✲ A m ) isdetermined by its image-dendrix if in addition the edges of the image-dendrix are“coloured” by units of C . These units are determined in the usual fashion by theelementary inert subobjects of the domains of A i + ✲ A i +1 , i = 0 , . . . , m −
1. Theresulting unit-coloured dendrix determines (up to isomorphism) the cocone withapix ([ m ] , A ) induced by the Segal core of C + . Therefore, this cocone is a colimitcocone for any C -tree, whence the density of the Segal core in ( C + ) in .Concerning extensionality, we have to construct a pushout of the cospan([ m ] , A • ) ✛ + ([1] , U + ✲ A ) > ✲ ([ n ] , B • )in C + . The inert morphism on the right induces in particular a commutative square A > ✲ B i +1 U + ✻ > ✲ B i + ✻ OMENT CATEGORIES AND OPERADS 29 in C . Observe that the total composition of the C -tree ([ m ] , A • ) equals the leftvertical morphism of the square. By extensionality of C we can thus push forwardalong U > ✲ B i the m -simplex A • . This defines a C -tree([ m + n − , B + ✲ · · · B i + ✲ B (1) i · · · + ✲ B ( m − i + ✲ B i +1 · · · + ✲ B n )realising the required pushout of the cospan. (cid:3) Remark . We call the pushout just constructed the result of inserting the C -tree ([ m ] , A • ) into the vertex of ([ n ] , B • ) represented by U > ✲ B i . This terminolgyis justified by the fact that after application of γ C + : C + → Γ + we get indeed theinsertion of the corresponding image-dendrices.3.4. Monadicity.
Our goal is to show that for each unital, extensional hyper-moment category C of finite type, the forgetful functor Oper C ( E ) → Coll C ( E ) is monadic . We achieve this by constructing a Feynman category F C in the sense ofKaufmann-Ward [24] such that C + -monoids (and hence C -operads) get identifiedwith F C -algebras so that we can apply monadicity of the latter.For our purpose, the following restricted definition is entirely sufficent, see [24]for a more general treatment. A Feynman category is a triple ( F , V , ι ) consistingof a permutative category F equipped with an embedding ι : V ֒ → F of a groupoid V such that the following two axioms hold:(i) the functor ι ⊗ : V ⊗ → F is an equivalence of perm. categories;(ii) the functor ( F ↓ V ) ⊗ → F ↓ F is an equivalence of perm. categories.A permutative category is a small, symmetric, strict monoidal category. Here V ⊗ denotes the free permutative category on V . Its objects are ordered (possiblyempty) tuples of objects of V and its morphisms ( v , . . . , v n ) → ( w , . . . , w n ) are( n + 1)-tuples ( σ, φ , . . . , φ n ) consisting of a permutation σ and φ i : v i ∼ = w σ ( i ) in V . Similarily, ( F ↓ V ) ⊗ denotes the free permutative category on F ↓ V .Morphisms belonging to
F ↓ V are called basic . V is the vertex-groupoid of F .The whole Feynman category can be reconstructed out of the basic morphisms, themonoidal structure and composition. The basic morphisms of a Feynman categoryform a groupoid-coloured symmetric operad in the sense of Petersen [31]. Definition 3.13.
For a unital, extensional hypermoment category C of finite typewe define a Feynman category F C with vertex-groupoid V C = ( C + ) unit .For each ( n + 1) -tuple of units ( i , . . . , i n ; i ) let F C ( i , . . . , i n ; i ) be the set of C -trees (ordered by γ C + ) for which the vertices have the units i , . . . , i n as domainsand for which the total composition of the active morphisms equals the unit i .In general, put F C ( i , . . . , i n ; j , . . . , j m ) = ` φ ∈ Γ act ( m,n ) ` i ∈ m F C ( g φ ( i ); i ) where g φ ( i ) stands for the subtuple of ( i , . . . , i n ) whose indices belong to φ ( i ) .Composition is defined by insertion of C -trees into vertices of C -trees, cf. Reamrk3.12. The monoidal structure is induced by concatenation of tuples. Two comments are in order. By the way F C is defined, axioms (i-ii) of a Feynmancategory are obvious because both functors are bijections. In particular, for giventuples ( i , . . . , i n ) , ( j , . . . , j m ) and φ ∈ Γ act ( m, n ), a morphism in the correspondingcomponent of F C decomposes uniquely as a tensor product of basic morphisms. Itsuffices thus to specify how to compose basic morphisms, and this is precisely doneby inserting a C -tree into the vertex of another C -tree. Proposition 3.14.
The category of F C -algebras is equivalent to the category of C + -monoids.Proof. The underlying object of an F C -algebra is a family X = ( X i ) i ∈ ( C + ) unit ofobjects of E equipped with Aut C + ( i )-actions. This is also the underlying object ofa C + -monoid. The F C -algebra structure consists of action-maps F C ( i , . . . , i n ; i ) ⊗ X i ⊗ · · · ⊗ X i n → X i compatible with composition. Each C -tree receives an essentially unique activemap from a unit i . For a C + -monoid X , the X -action of this map is preciselyof the form given above where ( i , . . . , i n ) are the domains of the vertices of thegiven C -tree. The X -action of a general active map between C -trees is a tensorproduct of X -actions of active maps with unital domain. Finally, any compositionof active maps between C -trees can be decomposed into a composition of insertionsof C -trees into vertices of C -trees. Therefore, the F C -algebra structure on X andthe C + -monoid structure on X determine each other. (cid:3) Theorem 3.15.
Let E be a cocomplete closed symmetric monoidal category, andlet C be a unital, extensional hypermoment category of finite type.Then the forgetful functor Oper C ( E ) → Coll C ( E ) is monadic.Proof. Combining Theorem 3.6 and Proposition 3.14 we get an equivalence betweenthe categories of C -operads and of F C -algebras. Under this equivalence (combinedwith the equivalence ( C + ) unit ≃ C iso ), the forgetful functor Oper C ( E ) → Coll C ( E )corresponds to the functor taking an F C -algebra to its underlying V C -module. Thelatter is monadic by [24, Theorems 1.5.3 and 1.5.6]. (cid:3) Remark . The preceding theorem also furnishes an explicit way of computingthe C -operad freely generated by a C -collection because the F C -algebra freely gen-erated by a V C -module is computable as a pointwise left Kan extension. The case C = Γ is of course well-known: F Γ is the Feynman category whose basic morphismsare rooted trees. One of the first places where the underlying coloured symmet-ric operad was described is [9]. Yet, Baez-Dolan [1] already mention this colouredsymmetric operad in their outline of the construction of opetopes. References [1] J. C. Baez and J. Dolan –
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Universit´e Cˆote d’Azur, Lab. J.-A. Dieudonn´e, Parc Valrose, 06108 Nice,France.