2-Segal sets and the Waldhausen construction
Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, Martina Rovelli, Claudia I. Scheimbauer
22-SEGAL SETS AND THE WALDHAUSENCONSTRUCTION
JULIA E. BERGNER, ANG´ELICA M. OSORNO, VIKTORIYA OZORNOVA,MARTINA ROVELLI, AND CLAUDIA I. SCHEIMBAUER
Abstract.
It is known by results of Dyckerhoff-Kapranov and of G´alvez–Carrillo-Kock-Tonks that the output of the Waldhausen S • -constructionhas a unital 2-Segal structure. Here, we prove that a certain S • -functordefines an equivalence between the category of augmented stable doublecategories and the category of unital 2-Segal sets. The inverse equiva-lence is described explicitly by a path construction. We illustrate theequivalence for the known examples of partial monoids, cobordism cat-egories with genus constraints and graph coalgebras. In [18], Waldhausen gave a definition of the algebraic K -theory of certaincategories using the S • -construction. The input categories for this construc-tion, now called Waldhausen categories, have specified cofibrations and weakequivalences, subject to some axioms, and generalize more classical notionssuch as exact categories. The essential step is to construct a simplicial spacewhose k th space of simplices is the classifying space of the groupoid of dia-grams of a certain shape.More recently, there have been many generalizations of this constructionusing several flavours of ( ∞ , unital2-Segal space by the former authors, and as a decomposition space by thelatter. Date : May 4, 2017.2010
Mathematics Subject Classification.
Key words and phrases. S • -construction, double categories,partial monoids, cobordism categories.The first-named author was partially supported by NSF CAREER award DMS-1352298. The second-named author was partially supported by a grant from the SimonsFoundation ( a r X i v : . [ m a t h . A T ] M a y lthough the two sets of authors come at the definition of unital 2-Segalspace from two different perspectives and are motivated by different exam-ples, it is significant that both groups of authors identify the output of the S • -construction as a key example. One can therefore ask whether all unital2-Segal spaces arise via an S • -construction for a suitably generalized inputcategory.In this paper, we restrict ourselves to the discrete case of unital 2-Segalsets, which are certain simplicial sets, rather than simplicial spaces. Webriefly describe this structure in this context; a more precise definition isgiven in the next section.A 1- Segal set is a simplicial set X such that the Segal maps X n → X × X · · · × X X (cid:124) (cid:123)(cid:122) (cid:125) n are isomorphisms for n ≥
2. This condition allows us to think of X ashaving an object set X , a morphism set X , and a composition which canbe defined by the span X × X X ← X → X , since the first arrow is an isomorphism. Indeed, a simplicial set is a 1-Segalset if and only if it is isomorphic to the nerve of a category.In contrast, a 2-Segal set is a simplicial set X such that certain maps X n → X × X · · · × X X (cid:124) (cid:123)(cid:122) (cid:125) n − are isomorphisms for n ≥
3. In this setting, we still have an object set X and a morphism set X , but we no longer have composition of all morphisms,since the first map in the span X × X X ← X → X is no longer necessarily invertible. However, we can think of a 2-Segal setas having a multi-valued composition, where an element of X × X X islifted to a preimage in X , which is in turn sent to its image in X . Thus,two potentially composable morphisms could have no composite at all (ifthe preimage in X is empty) or multiple composites (if the preimage hasmultiple elements). The invertibility of the 2-Segal maps given above is usedto prove that this multi-valued composition is associative. We think of thisstructure as that of a multi-valued category .To understand what the 2-Segal condition does, let us look more preciselyat how the maps are defined. Unlike the case of 1-Segal maps, where for every n ≥ n + 1)-gons for n ≥
3. For example, if X is a simplicial set, thetwo triangulations T and 0 123 T of the square induce two different maps X → X × X X ; indeed, differentface maps are used to define the pullbacks corresponding to the two triangu-lations. The fact that X is isomorphic to both pullbacks gives associativityof the partially defined composition. We specify these maps more preciselyin the following section.We also restrict ourselves to unital S • -construction in this context. When appliedto an exact category, the output of the S • -construction is a simplicial spaceobtained by taking the geometric realization of the groupoid of diagrams of acertain shape in every degree. It is a result of both Dyckerhoff-Kapranov andG´alvez–Carrillo-Kock-Tonks that this simplicial space is 2-Segal. Roughlyspeaking, a 3-simplex arises from a cartesian square in the exact category,and one of the two 2-Segal maps in degree 3 extracts the cospanof this square. As the cartesian square is determined up to isomorphism byits cospan, the corresponding 2-Segal map is a weak equivalence.To get a 2-Segal set, a naive guess would be to take the set of all diagramsof the required shape, rather than the classifying space of the groupoid ofsuch diagrams of a particular shape. Although this construction produces asimplicial set, it is not 2-Segal. The obstruction boils down to the fact thatthe cartesian square that completes a cospan in an exact category is onlydetermined up to isomorphism. Since this square is not defined uniquely,the 2-Segal map fails to be an isomorphism.In this paper, we identify the optimal amount of structure so that adiscrete version of the S • -construction can be defined and is a 2-Segal set.Although exact categories do not fit into this discrete context, they inspirethe structure and the properties we are looking for. For instance, the inputobject should have a collection of distinguished squares with the propertythat every cospan can be completed to a distinguished square in a unique ay. These ideas lead to the notion of stable pointed double category , whichwe now briefly explain.A double category is a category internal to categories. More informally,it consists of the following data, subject to some axioms: a set of objects;two different morphism sets which we suggestively call “horizontal” and“vertical” morphisms; and “squares”, which have “horizontal” source andtarget morphisms and “vertical” source and target morphisms. We depict asquare by .To define a simplicial object via an S • -construction, we need the input tobe pointed . In the framework of exact categories, we assume a zero object,but in the double category setup we only ask for an object ∗ which is initialfor the horizontal category and terminal for the vertical category. For sucha pointed double category D we let S n ( D ) be the set of diagrams of the form ∗ a a a · · · a n ∗ a a · · · a n ∗ ∗ a ( n − n ∗ . ... ... As n varies, we obtain a simplicial set whose face maps are given by deletinga row and a column, and composing appropriately.If the pointed double category D is stable , meaning that any square isuniquely determined by the span composed by its horizontal and verticalsources, and, simultaneously, by the cospan composed by its horizontal andvertical targets, then S • ( D ) is a 2-Segal set.Observe that S ( D ) = {∗} ; we call a 2-Segal set with this property re-duced . To obtain 2-Segal sets which are not reduced, we replace the singletonset {∗} by a subset of objects called the augmentation . When taking the S • -construction, we require the elements along the diagonal to be in theaugmentation set. he definitions of double categories and the conditions on them whichwe require can be found in Section 3, and the S • -construction is describedexplicitly in Section 4. The following is the statement of our main result,Theorem 6.1. Main Theorem.
The generalized S • -construction defines an equivalence ofcategories between the category of augmented stable double categories andthe category of unital 2-Segal sets. The inverse functor can also be described explicitly via a path construc-tion or d´ecalage functor, as we explain in Section 5. In this paper, weillustrate the equivalence of the above theorem by three examples of 2-Segalsets which do not arise naturally from the ordinary S • -construction. Thefirst one encodes the structure of Segal’s partial monoids [16]; the secondone is borrowed from work of the fifth-named author and Valentino [14],and encodes 2-dimensional cobordisms with genus constraints. The thirdone is a 2-Segal set associated to a graph, and is a more basic version of a2-Segal space encoding the combinatorics of graphs, related to the Schmittcoalgebra [10], [15]. We describe these 2-Segal sets in Section 2. In Sec-tion 7, we return to these examples and give an explicit description of theassociated augmented stable double category that results from applying thepath construction functor to each.As mentioned earlier, the S • -construction of an exact category does notfit into the framework described in this paper, because it fails to be stablein the discrete setting. In a future paper, we will establish an equivalenceof homotopy theories between unital 2-Segal spaces and double Segal spaceswhich are augmented and stable in a sense which appropriately generalizesthe usage in the current paper. The latter give an appropriate model for ahomotopical version of augmented stable double categories.Such structures are likely to arise not only from exact categories, butalso from stable ( ∞ , S • -construction has beenalready defined in [1], [2], and [7]. We expect our result to recover theseknown constructions. Acknowledgements.
We would like to thank the organizers of the Womenin Topology II Workshop and the Banff International Research Station forproviding a wonderful opportunity for collaborative research. Conversationswith T. Dyckerhoff, I. G´alvez–Carrillo, and A. Tonks were helpful. The fifth-named author would like to thank A. Valentino for conversations which ledto the example of cobordisms with genus constraints. We also thank thereferee for a detailed report which led to many improvements to this paper.1.
Some background on 2-Segal sets
Before introducing 2-Segal sets, we recall the more familiar definition of1-Segal sets. efinition 1.1. A simplicial set X is a if, for all n ≥
2, the map X n → X × X · · · × X X (cid:124) (cid:123)(cid:122) (cid:125) n , induced by the maps [1] ∼ = { i ≤ i + 1 } (cid:44) → [ n ] in the category ∆, for all0 ≤ i < n , is a bijection. Remark . When unspecified, pullbacks of X over X follow the conven-tion X × X X := lim (cid:104) X d −→ X d ←− X (cid:105) . The maps in Definition 1.1 can be defined for any simplicial object, notjust for simplicial sets, and of particular interest has been the case of simpli-cial spaces, in which these maps are required to be weak equivalences ratherthan isomorphisms, and the pullbacks are taken to be homotopy pullbacks.Rezk calls such simplicial spaces
Segal spaces in [13], taking the name fromsimilar maps used by Segal in his work on Γ-spaces in [17]. Dyckerhoff andKapranov use the term to distinguish them from more general n -Segal spaces, and we follow their usage here. While we restrict ourselvesto the case of simplicial sets in this paper, and defer homotopical variantsto later work, the following proposition points to the importance of 1-Segalspaces in defining homotopical categories. Proposition 1.3.
A simplicial set is 1-Segal if and only if it is isomorphicto the nerve of a category.Remark . Given a 1-Segal set X , the corresponding category can be recov-ered using the fundamental category functor τ . This functor is left adjointto the nerve functor, and moreover, it is a left inverse (i.e., the counit is anisomorphism); see [8, § II.4] for more details. For a general simplicial set X ,the set of objects of τ X is X , and the morphisms are given by equivalenceclasses of strings of elements in X , where the equivalence relation is gen-erated by elements of X . If X is a 1-Segal set, every equivalence class isuniquely represented by an element in X . Thus, in this particular case, thesets of objects and morphisms of τ X are given by X and X , respectively,and the source, target, and identity maps are provided by d , d , and s .Finally, composition is given by the composite X × X X d ,d ) − −−−−−−→ ∼ = X d −→ X . Here, we use the fact that X is 1-Segal, and hence that the map ( d , d ) isan isomorphism, to obtain a single composition map X × X X → X . Wefrequently define composition maps via such diagrams in this paper.We consider the following 2-dimensional generalization of 1-Segal sets asgiven by [4]. efinition 1.5. A is a simplicial set such that, for every n ≥ T of a regular ( n + 1)-gon, the induced map X n → X × X · · · × X X (cid:124) (cid:123)(cid:122) (cid:125) n − , which we call the T - Segal map , is an isomorphism of sets.Let us explain further how these triangulations of polygons induce suchmaps. To do so, we make use of the following notation. If S is a subsetof { , , . . . , n } , and X is a simplicial set, then we denote by X S the set of | S | -simplices of X . This notation is useful to specify face maps; for examplewe can denote d : X → X by X { , , } → X { , } .For n ≥
3, consider a regular ( n + 1)-gon with a cyclic labelling of itsvertices by the set { , , . . . , n } . A triangulation of such an ( n + 1)-gondetermines n − { , . . . , n } with exactly three elements (the tri-angles), and the iterated pullback in the T -Segal map is induced by takingtriangles which agree along a 1-dimensional face (a two-element subset).For example, when n = 3, the two triangulations0 123 T and 0 123 T of the square determine the two diagrams X X { , , } X { , , } X { , } d d d d and X X { , , } X { , , } X { , } , d d d d which in turn give two maps f T : X −→ X { , , } × X { , } X { , , } and f T : X −→ X { , , } × X { , } X { , , } . The following result follows from the definition of 2-Segal set.
Proposition 1.6.
Let X be a simplicial set and Y a 2-Segal set. (1) A simplicial map g : X → Y is uniquely determined by g i for i =0 , , . Given g i : X i → Y i for i = 0 , , , compatible with the simplicialstructure, there exists an extension to a simplicial map g : X → Y . (3) If X is also a 2-Segal set, then g is an isomorphism of simplicialsets if and only if g i is an isomorphism for i = 0 , , .Proof. Given n ≥
3, consider a triangulation T of the ( n + 1)-gon. Thencommutativity of the diagram X n g n (cid:47) (cid:47) f T (cid:15) (cid:15) Y n ∼ = f T (cid:15) (cid:15) X × X · · · × X X g ×···× g (cid:47) (cid:47) Y × Y · · · × Y Y , where the vertical arrows are the T -Segal maps, shows that g n is completelydetermined by g and g , proving (1). The same diagram proves (3): if thetwo T -Segal maps are isomorphisms and g and g are isomorphisms, thenthe bottom map is also an isomorphism, and hence so is g n .To prove (2), we first prove that for n ≥ T and T (cid:48) of the ( n + 1)-gon, the diagram X × X · · · × X X g ×···× g (cid:47) (cid:47) Y × Y · · · × Y Y ∼ = f − T (cid:39) (cid:39) X n f T (cid:48) (cid:39) (cid:39) f T (cid:55) (cid:55) Y n X × X · · · × X X g ×···× g (cid:47) (cid:47) Y × Y · · · × Y Y ∼ = f − T (cid:48) (cid:55) (cid:55) commutes. When n = 3 there are only two triangulations, and the statementis true since we are given g , which is equal to both composites as shownin the diagram above. For n ≥
4, if two triangulations differ only by oneedge, so that we can obtain one from the other by changing the diagonal ofa sub-quadrilateral in the ( n + 1)-gon, the result follows from the fact thatthe 2-Segal maps f T and f T (cid:48) factor as X × X · · · × X X X n f T (cid:48) (cid:43) (cid:43) f T (cid:51) (cid:51) (cid:47) (cid:47) X × X · · · × X X × X · · · × X X ×···× f T ×···× id (cid:79) (cid:79) id ×···× f T (cid:48) ×···× id (cid:15) (cid:15) X × X · · · × X X , here T and T (cid:48) are the two triangulations of the sub-quadrilateral specifiedabove. Since any two triangulations can be related by a sequence of suchone-edge flips, the result follows for arbitrary triangulations of the ( n + 1)-gon.We can thus define g n as the composite f − T ◦ ( g × · · · × g ) ◦ f T forany triangulation T . It remains to show that these maps are compatiblewith the simplicial structure. We prove the compatibility with d n : X n → X n − ; the other faces and the degeneracies are proved similarly. Let T be atriangulation of the ( n + 1)-gon that contains the triangle { , n − , n } , andlet T (cid:48) be the corresponding triangulation of the n -gon, obtained by removingthat precise triangle. Then we have the commutative diagram X n f T (cid:47) (cid:47) d n (cid:15) (cid:15) n − (cid:122) (cid:125)(cid:124) (cid:123) X × X · · · × X X g ×···× g (cid:47) (cid:47) (cid:15) (cid:15) n − (cid:122) (cid:125)(cid:124) (cid:123) Y × Y · · · × Y Y (cid:15) (cid:15) ∼ = f − T (cid:47) (cid:47) Y nd n (cid:15) (cid:15) X n − f T (cid:48) (cid:47) (cid:47) X × X · · · × X X (cid:124) (cid:123)(cid:122) (cid:125) n − g ×···× g (cid:47) (cid:47) Y × Y · · · × Y Y (cid:124) (cid:123)(cid:122) (cid:125) n − ∼ = f − T (cid:48) (cid:47) (cid:47) Y n − , where the two vertical maps in the middle are projections onto all but oneof the factors, corresponding to the triangle that was removed. (cid:3) We thus obtain the following consequence; for more details on coskeletalsimplicial sets, see, for example [11, § IV.3.2].
Corollary 1.7. If Y is a 2-Segal set then it is 3-coskeletal.Proof. Let ∆ ≤ denote the truncation of ∆ containing only the objects [ i ]for i ≤
3, and let s S et ≤ denote the category of functors ∆ op ≤ → S et . Notethat (1) and (2) in Proposition 1.6 imply that for X any simplicial set and Y a 2-Segal set, the mapHom s S et ( X, Y ) −→ Hom s S et ≤ (tr X, tr Y )is an isomorphism. Thus any 2-Segal set Y is 3-coskeletal. (cid:3) Remark . For an interval [ x , x n ], we can consider a “triangulation” T given by vertices { x < x < · · · < x n } , x x x · · · x n − x n . The combinatorics of the subdivision induce the usual 1-Segal maps X n → X × X · · · × X X . Thus, we can see how 2-Segal sets give a “higher-dimensional” generalizationof 1-Segal sets. ne can also check that any 1-Segal set is also 2-Segal; a proof can befound in [4].We now consider some additional properties on 2-Segal sets. Definition 1.9.
A 2-Segal set X is reduced if X consists of a single point.The following definition is taken from [4]. Definition 1.10.
A 2-Segal set is unital if for all n ≥ ≤ i ≤ n − X n − X X n X α i s i s β i is a pullback, where s and s i are degeneracy maps and the maps α i , β i areinduced by the following maps: α i : [0] → [ n − (cid:55)→ i,β i : [1] → [ n ]0 (cid:55)→ i (cid:55)→ i + 1 . The conditions for unitality of a 2-Segal set can be reduced by the fol-lowing lemma. The proof is analogous to that of a similar reduction in[9].
Lemma 1.11.
A 2-Segal set X is unital if and only if X X X X d s s d and X X X X d s s d are pullback diagrams. Notation 1.12.
Let U S eg and U S eg ∗ denote the full subcategories of s S et consisting of unital 2-Segal sets and reduced unital 2-Segal sets, re-spectively.A useful way to check that a simplicial set is 2-Segal is the Path SpaceCriterion, which can be found in [4] and in a similar form in [9].Recall the endofunctors of ∆ which are used to define the path spaceconstruction, i : ∆ → ∆[ n ] (cid:55)→ [0] ∗ [ n ] and f : ∆ → ∆[ n ] (cid:55)→ [ n ] ∗ [0] , here ∗ denotes the join operation, which for linear posets is simply given byconcatenation. Here, the functor names are meant to suggest adjoining aninitial and final object, respectively. Note that there are natural transforma-tions i ⇒ id ∆ and f ⇒ id ∆ induced by the maps δ : [ n ] → [0] ∗ [ n ] = [ n + 1]and δ n +1 : [ n ] → [ n ] ∗ [0] = [ n + 1], respectively. Definition 1.13 ([4]) . Given a simplicial set X , its path spaces are thesimplicial sets P (cid:67) ( X ) = X ◦ i op and P (cid:66) ( X ) = X ◦ f op .The natural transformations above induce maps of simplicial sets d : P (cid:67) ( X ) → X and d top : P (cid:66) ( X ) → X that are natural in X . These simplicial sets are also often called d´ecalages ,for example in [9].The following Path Space Criterion relates 1-Segal sets and 2-Segal sets. Theorem 1.14 ([4], [9]) . A simplicial set X is -Segal if and only if its pathspaces P (cid:67) X and P (cid:66) X are -Segal sets. Three examples
In this section, we give explicit descriptions of three examples of 2-Segalsets. The first is that of partial monoids; we follow the treatment of Segalin [16].
Example . A partial monoid is a set M together with a subset M ⊆ M × M which is the domain of an associative multiplication. In otherwords, there is a map · : M → M such that(1) for all m, m (cid:48) , m (cid:48)(cid:48) ∈ M , we have that ( m · m (cid:48) ) · m (cid:48)(cid:48) is defined if anonly if m · ( m (cid:48) · m (cid:48)(cid:48) ) is defined, and if they are both defined, then wehave associativity ( m · m (cid:48) ) · m (cid:48)(cid:48) = m · ( m (cid:48) · m (cid:48)(cid:48) ) , (2) there is a unit ∈ M such that for every m ∈ M , we have that(1 , m ) ∈ M and ( m, ∈ M , and 1 · m = m · m .In [16], Segal defined the nerve of a partial monoid as the following sim-plicial set. Let M = { } , and for k ≥
1, let M k ⊆ M × k be the subsetof composable k -tuples , which are elements ( m , . . . , m k ) ∈ M × k such that( m · · · m i , m i +1 ) ∈ M for every 1 ≤ i < k . The face maps are given bycomposition, and the degeneracy maps are defined using insertion of theunit 1. More precisely, for 0 ≤ i ≤ k , d i : ( m , . . . , m k ) (cid:55)−→ ( m , . . . , m k ) , i = 0 , ( m , . . . , m i · m i +1 , . . . , m k ) , < i < k, ( m , . . . , m k − ) , i = k ; s i : ( m , . . . , m k ) (cid:55)−→ ( m , . . . , m i , , m i +1 , . . . , m k ) . e claim that M • is a 2-Segal set. Indeed, let T be any triangulation of apolygon with n vertices. We need to check that the induced T -Segal map f T : M n → M × M · · · × M M (cid:124) (cid:123)(cid:122) (cid:125) n − is a bijection. The maps used in the fiber product on the right-hand sideare face maps.Let us first illustrate this bijection for an example. Consider the followingtriangulation of a pentagon: 0123 4.An element in M is a composable 4-tuple ( m , m , m , m ), i.e., we havethat ( m , m ) , ( m m , m ) , and ( m m m , m ) ∈ M . The T -Segal maplands in the fiber product M d × d M M d × d M M , and it sends ( m , m , m , m ) to(( m , m ) , ( m · m , m · m ) , ( m , m )) . An arbitrary element in the fiber product above is a triple( m , m ) , ( n , n ) , ( m , m )of elements in M such that n = m · m and n = m · m . We can thinkof such an element as a decoration of the triangulation: m m m m
4. ( m · m ) · ( m · m ) m · m m · m Since ( n , n ) = ( m · m , m · m ) ∈ M , we have that ( m · m ) · ( m · m ) iswell-defined and by associativity, ( m · m · m , m ) ∈ M . In particular, allother necessary products are well-defined and therefore ( m , m , m , m ) ∈ M . he essential ingredient of this argument is to relate an element in thefiber product appearing in the T -Segal map with a decoration of the trian-gulation. In particular, such an element determines the labels m , . . . , m n decorating the outer edges of the n -gon except for the last one (the 0 n edge).The diagonals in the interior are decorated by iterated products of some ofthese elements. Finally, the 0 n edge is decorated by an iterated product ofall of the elements; it is a classical result going back to Catalan [3] that tri-angulations of an ( n +1)-gon are in one-to-one correspondence with the waysof bracketing a product of n elements. Associativity of the multiplicationthen implies that ( m , . . . , m n ) ∈ M n .Moreover, the 2-Segal set M • is unital. The map β i : M n → M of Def-inition 1.10 sends ( m , . . . , m n ) ∈ M n to m i +1 . Thus, an element of thepullback of the diagram M s (cid:15) (cid:15) M n β i (cid:47) (cid:47) M is an element of M n of the form ( m , . . . , m i , , m i +2 , . . . , m n ). The mapfrom M n − to the pullback induced by the universal property, which sends( m , . . . , m n − ) to ( m , . . . , m i , , m i +1 , . . . , m n − ), is then an isomorphism.The second example is that of cobordisms with a genus constraint, whichis taken from and treated in more detail in work of the fifth-named authorand Valentino in [14]. Example . Fix a non-negative integer g , and consider 2-dimensional cobor-disms with the constraint that its genus is less than or equal to g . Followingthe definition of (the nerve of) the usual 2-dimensional cobordism category,we consider the following simplicial set 2Cob ≤ g . • Let the elements in (2Cob ≤ g ) be 1-dimensional closed manifolds,which can be depicted by · · · and are just disjoint unions of circles. • Let the elements of (2Cob ≤ g ) k be diffeomorphism classes of 2-dimen-sional cobordisms Σ between 1-dimensional closed manifolds ∂ in Σand ∂ out Σ with genus less than or equal to g , together with a de-composition thereof into k cobordisms Σ , . . . , Σ k . Here, a diffeo-morphism of such decomposed cobordisms must restrict to the indi-vidual composed cobordisms Σ i for 1 ≤ i ≤ k . We write (Σ , . . . , Σ k )for an element in (2Cob ≤ g ) k , since the individual cobordisms fullydetermine the k -simplex. or example, if g = 0, the left picture is allowed as a 3-simplex of 2Cob ≤ ,whereas the second one is not: Σ Σ Σ Σ Σ Σ . Observe that 2Cob ≤ g is not the nerve of a category, because not all pairswith compatible outgoing and incoming boundary components compose, asillustrated by the picture above. However, it is a unital 2-Segal set which is asimplicial subset of the nerve of the usual 2-dimensional cobordism category2Cob.Our next example is inspired by the example of the 2-Segal space of graphsdescribed by G´alvez–Carrillo, Kock, and Tonks in [10], from which one canobtain the Hopf algebra of graphs. Our example instead looks at a 2-Segalset corresponding to a single graph. Example . Let G be a graph consisting of a set v ( G ) of vertices of G anda set of edges between vertices. We associate to G a simplicial set X asfollows.(1) The set X has a single element which we denote by ∅ .(2) The set X is the set of all subgraphs of G .(3) Any X n has elements ( H ; S , . . . , S n ) where H is a subgraph of G and the sets S , . . . , S n form a partition of the set v ( H ) of verticesinto n disjoint (but possibly empty) sets.(4) The face maps d i : X n → X n − are defined in the following way.(a) If i = 0, then d ( H ; S , . . . , S n ) = ( H (0) ; S , . . . , S n )where H (0) denotes the full subgraph of H on the vertices v ( H ) \ S .(b) If i = n , then d n ( H ; S , . . . , S n ) = ( H ( n ) ; S , . . . , S n − )where H ( n ) denotes the full subgraph of H on the vertices v ( H ) \ S n . c) If 0 < i < n , then d i ( H ; S , . . . , S n ) = ( H ; S , . . . S i − , S i ∪ S i +1 , S i +2 , . . . , S n ) . (5) The degeneracy maps s i : X n → X n +1 are given by s i ( H ; S , . . . , S n ) = ( H ; S , . . . S i , ∅ , S i +1 , . . . , S n ) . Observe that condition (2) is actually encompassed by condition (3), ifwe regard a graph as being partitioned into a single set. We use this unin-teresting partition in the depictions of particular examples that follow.One can check that this simplicial set is 2-Segal but not 1-Segal. Let usinvestigate a specific example. Consider the graph G : a b c. Then we can depict the set of 1-simplices as: X : ∅ a b c a b a b a cb c b c a b ca b c a b ca b c. We do not list all the 2-simplices, but illustrate with some examples offace maps. We illustrate partitions by colored circles; we use blue for thefirst element of the partition, orange for the second, and, for 3-simplices,green for the third. In most examples the partitions are ordered from leftto right. First, we show the effect of all the face maps on a representativeelement of X : b cb c a b c a.d d d Likewise, we depict the degeneracies of a particular 1-simplex as follows: a b ∅ a b a b ∅ .s s More interestingly, we have the following face maps of a 3-simplex of X : a b cb c a b c a b c a b.d d d d We can start to see from these face maps how one might reconstruct a3-simplex from its faces.Indeed, the fact that the associated simplicial set X is 2-Segal can beillustrated by the diagrams a b cb c a b cb cd d d d nd a b ca b c a ba b.d d d d However, X is not 1-Segal, as can be illustrated by the diagrams a ba bd d and a ba b.d d In fact, this example shows that, unlike for the other two examples givenin this section, the 1-Segal map X → X × X X is not injective.One can check, however, that these 2-Segal sets arising from graphs arealways unital. 3. Augmented stable double categories
Our aim is to give a description of unital 2-Segal sets in terms of cate-gorical structures to which we can apply some version of Waldhausen S • -construction. Here, we work with structures that have two different kinds ofmorphisms on the same set of objects, which are better described in termsof double categories. Double categories were first defined by Ehresmann [5],and good introductory accounts can be found in [6] and [12]. We begin byrecalling the definition. Definition 3.1. A (small) double category is an internal category in thecategory of small categories. Remark . Roughly speaking, being an internal category in the categoryof small categories means that a double category consists of a “category ofobjects” and a “category of morphisms”. However, this definition obscuresthe symmetry between the two kinds of morphisms. Hence, it is useful tounpack this definition further to see that a double category D consists of setsof objects Ob D , horizontal morphisms Hor D , vertical morphisms V er D , nd squares Sq D , which are connected by various source, target, identityand composition maps.In particular, ( Ob D , Hor D ) and ( Ob D , V er D ) form categories which wedenote by H or D and V er D , respectively. The subscript 0 indicates thateach of these categories can be considered as a category of objects in acategory internal to categories, as we explain below. We denote horizontaland vertical morphisms by and , respectively. Squares are depicted by diagrams of the following form: kfj gα .There are horizontal and vertical source and target maps s h , t h : Sq D →
V er D and s v , t v : Sq D →
Hor D given by, for example, s h ( α ) = j . Note that the horizontal source andtarget of a square are vertical morphisms, and similarly the vertical sourceand target of a square are horizontal morphisms. In a double category wehave horizontal composition of squares ◦ h : Sq D ×
V er D Sq D → Sq D and vertical composition of squares ◦ v : Sq D ×
Hor D Sq D → Sq D . These compositions are compatible with the compositions in H or D and V er D via source and target maps. The compatibility allows depicting hor-izontal and vertical composition as and There are also horizontal and vertical identity squares, respectively, whichcan be depicted by the following diagrams: j id h j id h id hj and .id v f id v f id vf This data is further subject to the following axioms: • associativity of the compositions ◦ h and ◦ v ; • unitality of ◦ h and ◦ v : the identity squares id h and id v act as theidentity for horizontal and vertical composition of squares, respec-tively; and • interchange law: given squares as in the following diagram, the orderof composition does not matter, i.e., we can first compose eitherhorizontally or vertically: .From a double category D we can extract two categories H or D and V er D corresponding to horizontal and vertical composition of squares, re-spectively. For example, the category H or D has V er D as the set of objectsand Sq D as the set of morphisms, with source, target, identity, and compo-sition given by s h , t h , id h , and ◦ h , respectively. In fact, the categories V er D and V er D are the “categories of objects” and “categories of morphisms”,respectively, of the category internal to categories D . Similarly, the pair( H or D , H or D ) also forms a category internal to categories, which is the transpose of D . he following diagram, adapted from [12], gives a useful depiction of somethe sets and categories that are associated to a double category and whichhave been described above. Ob D (cid:47) (cid:47) (cid:15) (cid:15) Hor D t h (cid:111) (cid:111) s h (cid:111) (cid:111) (cid:15) (cid:15) H or D (cid:15) (cid:15) V er D (cid:47) (cid:47) t v (cid:79) (cid:79) s v (cid:79) (cid:79) Sq D t h (cid:111) (cid:111) s h (cid:111) (cid:111) t v (cid:79) (cid:79) s v (cid:79) (cid:79) H or D (cid:79) (cid:79) (cid:79) (cid:79) V er D (cid:47) (cid:47) V er D (cid:111) (cid:111) (cid:111) (cid:111) Example . If E is an exact category, the full subcategories of admissiblemonomorphisms and admissible epimorphisms assemble to the data of adouble category, where the collection of distinguished squares is given by allcommutative squares which are simultaneously pullbacks and pushouts.We can also consider appropriate functors between double categories. Definition 3.4. A double functor between double categories is an internalfunctor.In other words, a double functor consists of an assignment of objects,horizontal and vertical morphisms, and squares, all of which are compatiblewith all source, target, composition, and unit maps.To define a meaningful version of the Waldhausen S • -construction, weneed to impose further conditions on double categories. Recall that anexact category has a zero object, namely an object which is both initial andterminal. Since categories with zero objects are often referred to as pointed,we use this terminology for our generalization to double categories. Definition 3.5.
A double category D is pointed if it is equipped with afixed distinguished object ∗ which is both initial in H or D and terminal in V er D . A double functor F : D → D (cid:48) between pointed double categories is pointed if it sends the distinguished object of D to the distinguished objectof D (cid:48) .In order to obtain all 2-Segal sets, and not just those with a single 0-simplex, we will have need of a more general notion than pointedness. Definition 3.6. An augmentation of a double category D consists of a setof objects A satisfying the condition that for every object d of D there areunique morphisms a (cid:26) d in Hor D and d (cid:16) a (cid:48) in V er D such that a and a (cid:48) are in A .An augmented double category is a double category equipped with a choiceof augmentation. A double functor between augmented double categories is augmented if objects in the augmentation of the source are sent to objectsin the augmentation of the target. emark . Note that if the augmentation set is a singleton, so A = {∗} ,the definition reduces to being pointed by the single object ∗ .The following result is a reformulation of the notion of augmentation incategorical terms. Proposition 3.8.
A subset A of Ob D is an augmentation for the doublecategory D if and only if there exist maps f : Ob D −→
Hor D g : Ob D −→
V er D δ h : Ob D −→
A δ v : Ob D −→ A, such that t h ◦ f = id and s v ◦ g = id , and the two diagrams Ob D δ h (cid:47) (cid:47) f (cid:15) (cid:15) A (cid:127) (cid:95) (cid:15) (cid:15) Ob D δ v (cid:47) (cid:47) g (cid:15) (cid:15) A (cid:127) (cid:95) (cid:15) (cid:15) and Hor D s h (cid:47) (cid:47) Ob D V er D t v (cid:47) (cid:47) Ob D commute and are pullback diagrams.Equivalently, A is an augmentation for D if and only if the maps t h ◦ pr : Hor
D × Ob D A → Ob D and s h ◦ pr : V er
D × Ob D A → Ob D are bijections.Remark . In particular, the maps f and δ h of Proposition 3.8 make thenerve of the category H or D into an augmented simplicial set with an extradegeneracy, more precisely, with a backward contracting homotopy, A Ob D Hor D Hor
D × Ob D Hor D , δ h f f × id · · · where f × id : Hor
D ∼ = Ob D × Ob D Hor
D →
Hor
D × Ob D Hor D , andsimilarly for the other extra degeneracy maps.Similarly, g and δ v make the nerve of the category V er D into an aug-mented simplicial set with a forward contracting homotopy given by g , A Ob D V er D V er
D × Ob D V er D . δ v g id × g · · · In the definition of the S • -construction for exact categories, certain squaresare required to be both pullbacks and pushouts. In other words, such asquare is determined, up to isomorphism, by its span; it is similarly de-termined by its cospan. In a double category, this condition does not make ense, since in general the horizontal and vertical morphisms in a double cat-egory need not be morphisms in a common category, so the span or cospanof a square may not be given by a diagram in a category. Furthermore, pull-backs and pushouts are only determined up to isomorphism, but we needuniqueness on the nose in the discrete setting. Definition 3.10.
A double category is stable if every square is uniquely de-termined by its span of source morphisms and, independently, by its cospanof target morphisms. More precisely, we require the maps( s h , s v ) : Sq D →
V er
D × Ob D Hor D ( t h , t v ) : Sq D →
V er
D × Ob D Hor D depicted by andto be bijections. Remark . The double category associated to an exact category is notstable, except for trivial examples. Indeed, a cartesian square is determinedby the corresponding span only up to isomorphism.An important consequence of stability, which we now show, is that theset of squares and their horizontal and vertical compositions are essentiallydetermined by the rest of the data.Let H and V be categories with the same set of objects Ob , and Sq a settogether with maps s h , t h : Sq → mor V and s v , t v : Sq → mor H , which we call horizontal and vertical source and target maps, respectively.As suggested by the name, the set Sq is a proposed set of squares for adouble category. Assume that these maps are compatible with the sourceand target maps in H and V , as in the definition of a double category (seeRemark 3.2). For example, s H s v = s V s h . In other words, we are given someof the data and axioms for a double category as detailed in Remark 3.2,missing precisely horizontal and vertical compositions of squares and exis-tence of horizontal and vertical identity squares. Furthermore, assume thatthe stability condition holds, i.e., the two maps(3.12) mor H × Ob mor V Sq ( t v ,t h ) (cid:111) (cid:111) ( s h ,s v ) (cid:47) (cid:47) mor V × Ob mor H are bijections. hen there are two horizontal “compositions” of squares defined usingcomposition in H and two vertical “compositions” of squares defined usingcomposition in V , as follows.We define the first horizontal composition of squares using the bijectionin (3.12) given by ( s h , s v ) and the second using the bijection in (3.12) givenby ( t v , t h ):(3.13) ◦ h : Sq × mor V Sq (mor V × Ob mor H ) × Ob mor H mor V × Ob mor H Sq , ◦ h : Sq × mor V Sq mor H × Ob (mor H × Ob mor V )mor H × Ob mor V Sq . ( s h ,s v ) × s v id ×◦ H ( s h ,s v ) − ∼ = t v × ( t v ,t h ) ◦ H × id ( t v ,t h ) − ∼ = Pictorially, the two maps can be understood as follows: ∼ = ∼ =Similarly, we can define two vertical “compositions” ◦ v and ◦ v of squaresusing composition in V . Proposition 3.14.
Let H , V , Sq , s v , t v , s h , t h be as above. Assume furtherthat ◦ h = ◦ h and ◦ v = ◦ v . Then this data assembles into a stable double category, i.e., associativity andunitality of both compositions and the interchange law hold.Proof.
Note that, by construction, ◦ h is compatible with composition ofvertical sources and with horizontal source, while ◦ h is compatible withcomposition of vertical targets and with horizontal target. Thus, if ◦ h = ◦ h ,this assignment is compatible with all source and target maps, and we canthus think of it as a composition ◦ h that can be iterated. Since compositionin H is associative, stability implies associativity of composition, as bothways of associating correspond to squares with the same source span. A A priori, these maps are not fully compatible with source and target, and hence donot deserve to be called composition. imilar argument shows that ◦ v = ◦ v = ◦ v is an associative compositioncompatible with source and target maps.Stability also implies the interchange law since we are comparing squareswith the same source span. It remains to check the existence of horizontaland vertical identity squares. Given a vertical morphism m , by stability,there exist two squares as follows: mfk id α and . m lg id β Composing them, we obtain fk id lg idwhich is a square whose span of sources is the pair ( k, f ), so by stabilitythis square must be α , and therefore its vertical target g = g ◦ id is theidentity id and its horizontal target m must be l . Similarly, its cospan oftargets is ( g, l = m ), so by stability this square must be β , and its verticalsource f = id ◦ f must be the identity and its horizontal source k must be m . Finally, stability again shows that α = β acts as an identity for ◦ h .A similar argument shows the existence of vertical identities. (cid:3) Remark . Moreover, if C is a double category and D is a stable dou-ble category, when defining a double functor F : C → D , the assignment onsquares is uniquely determined by the assignment on vertical and horizon-tal morphisms. However, it is not true that to define a double functor, it isenough to define functors on the horizontal and vertical categories that coin-cide on objects; the assignment on squares still requires some compatibilitybetween the vertical and horizontal pieces.For double categories which are stable and augmented the necessary datacan be reduced even further.
Lemma 3.16.
Let D be an augmented stable double category. Then thereis a bijection between the set of horizontal arrows and the set of verticalarrows, i.e., Hor
D ∼ = V er D . Proof.
Given a horizontal morphism x (cid:26) y , there is a unique vertical mor-phism x (cid:16) a with a in the augmentation. Then, by stability, there is a nique square z.xa y The horizontal target of this square is a vertical morphism y (cid:16) z , which isuniquely determined by the original horizontal morphism x (cid:26) y . Note thatthe vertical target of the square is a horizontal morphism a (cid:26) z which iscompletely determined by z since a is in the augmentation. Conversely, ifwe start with a vertical morphism y (cid:16) z , we can use a dual argument torecover the horizontal morphism x (cid:26) y . (cid:3) Recall Theorem 1.14, which tells us that if C is a category, then P (cid:67) ( N C )and P (cid:66) ( N C ) are 1-Segal sets. The following proposition identifies the cate-gories associated to the 1-Segal sets P (cid:67) ( N H or D ) and P (cid:66) ( N V er D ). Proposition 3.17.
Let D be an augmented stable double category. Thenthere are isomorphisms of categories τ P (cid:67) ( N H or D ) ∼ = H or D and τ P (cid:66) ( N V er D ) ∼ = V er D such that the following diagrams with the adjoint isomorphisms commute: P (cid:67) ( N H or D ) N H or D P (cid:66) ( N V er D ) N V er D N H or D N V er D . ∼ = d s v ∼ = d top t h Moreover, these isomorphisms are natural in D .Proof. We prove the statement for the diagram on the left, the other onebeing similar. One can check that the set of objects of τ P (cid:67) ( N H or D ) isprecisely the set Hor D . Since the set of objects of H or D is the set V er D ,we have constructed in Lemma 3.16 a bijection between the object sets ofthese two categories. By inspection, this bijection commutes with the sourceand target maps to Ob D .Observe that the set of morphisms of τ P (cid:67) ( N H or D ) is( N H or D ) = Hor
D × Ob D Hor D , the set of pairs of composable horizontal morphisms. We use the samebijection from Lemma 3.16, together with stability, to obtain a bijection tothe sets of morphisms of H or ( D ), which is the set Sq D : Hor
D × Ob D Hor
D ∼ = V er
D × Ob D Hor D ∼ = ←− Sq D . Note that in the first fibered product, the maps to Ob D are ( t h , s h ) and inthe second one, the maps are ( s v , s h ), as required. Pictorially, the bijectionfrom Lemma 3.16 yields the left-hand vertical arrow in the following picture,and stability yields the displayed square: These bijections are compatible with source, target, and composition andthus we obtain an isomorphism of categories. Commutativity of the desireddiagram can be read off from the pictorial representation. By inspection,the constructions of the isomorphisms are natural in D . (cid:3) Notation 3.18.
We denote by DC at ∗ the category of pointed double cate-gories and pointed double functors. We similarly denote by DC at aug the cat-egory of augmented double categories and augmented double functors. Wedenote by DC at st ∗ and DC at staug the full subcategories of DC at ∗ and DC at aug ,respectively, consisting of stable objects.The aim of this paper is to prove that the categories DC at st ∗ and DC at staug are equivalent to U S eg ∗ and U S eg , respectively.We conclude this section with an example which is part of a family ofaugmented stable double categories which will be essential in our definitionof the generalized S • -construction. Example . Consider a double category, denoted by W , with objects ij for 0 ≤ i ≤ j ≤
2, and generated by the following non-identity horizontaland vertical morphisms and squares:00 0111 120222 . Since the only non-identity square is uniquely determined by its pair ofsources or by its pair of targets, W is a stable double category.Furthermore, the set A = { , , } defines an augmentation of W .For example, consider the object 02. There is a unique horizontal morphismwith target 02 and source in A , namely the composite of the horizontalmorphisms from 00 to 01 and from 01 to 02,00 02 . Similarly, the vertical morphism from 02 to 22 is the unique vertical mor-phism with source 02 and target in A . . The generalized Waldhausen construction
We are now ready to define our generalized S • -construction, which is afunctor S • : DC at aug → s S et , inspired by Waldhausen’s S • -construction [18].We begin by describing a cosimplicial object W • in augmented stabledouble categories generalizing Example 3.19. The S • -construction will bedefined by mapping this cosimplicial object into a given augmented stabledouble category. Definition 4.1.
Given any integer n ≥
0, define a double category W n asfollows. Its objects are pairs { ( i, j ) | ≤ i ≤ j ≤ n } ;for simplicity of notation, we simply write ij for the pair ( i, j ). A horizontalmorphism is a triple ( i, j, k ) with i ≤ j ≤ k , viewed as a map ij (cid:26) ik .Similarly, a vertical morphism is also a triple ( i, j, k ) with i ≤ k ≤ j , viewedas a map ij (cid:16) kj . The squares are exactly those of the form klijkj il with i ≤ k ≤ j ≤ l .The double category W n is augmented by the set A = { ii | ≤ i ≤ n } . Example . The double categories W n can be pictured for 0 ≤ n ≤ : 00 W : 00 0111 W : 00 0111 120222 W : 00 0111 120222 03132333 W : 00 0111 120222 03132333 0414243444 . In each case, the pictured arrows and squares generate the double category W n , and the augmentation consists of all elements on the diagonal. Remark . We can alternatively construct W n as a certain sub-doublecategory of a double category of functors as follows. First, recall the doublecategory of commutative squares in an arbitrary category C . The verticaland horizontal categories are both given by C , and the set of squares isexactly the set of commutative squares in C . Composition of squares andunits come from composition of morphisms in C and from identities in C ,respectively.For each n ≥
0, consider the double category arising in this way from thecategory Fun([1] , [ n ]). Since the categories Fun([1] , [ n ]) form a cosimplicialcategory as n varies, their respective double categories also assemble to forma cosimplicial double category.For each n ≥
0, let W n be the sub-double category of the double categoryof commutative squares in Fun([1] , [ n ]) consisting of: • all the objects of Fun([1] , [ n ]); • natural transformations which are the identity on the object 1 of [1]as vertical morphisms; • natural transformations which are the identity on the object 0 of [1]as horizontal morphisms; and • all commutative squares between these morphisms.One can check that these properties do indeed define a double category.Moreover, since the cosimplicial structure comes from the post-compositionof morphisms [1] → [ n ] with arbitrary morphisms in ∆, it preserves the ertical and horizontal subcategories described above, thus making W • into acosimplicial double category. Each double category W n has an augmentationgiven by the constant functors [1] → [ n ]; note that the cosimplicial structuremaps preserve the augmentation. Lemma 4.4.
The collection W • is a cosimplicial object in augmented stabledouble categories.Proof. We have shown that each W n is an augmented double category; itremains to show that it is stable. If we have a span ijkj il and want to produce a square xy,ijkj il it is necessary that y = l for the right-hand side map to be a verticalmorphism, and that x = k for the lower map to be a horizontal morphism.Now there exists a unique square of the necessary form, namely kl,ijkj il since by properties of the original span we know that i ≤ k ≤ j ≤ l . Theargument for the cospan case is analogous. (cid:3) There are two other cosimplicial objects in double categories which willuse later on; by construction one governs the horizontal and the other thevertical category of a double category. We will explain the horizontal case,the vertical one is similar.
Definition 4.5.
For every n ≥
0, let H n be the (stable) double categorywhose object set is { , , . . . , n } , whose horizontal category is [ n ], whosevertical category is discrete, i.e., has no non-identity morphisms, and whosesquares are only the identity squares. Similarly, let V n be the (stable) doublecategory given by switching the horizontal and vertical directions in H n .In what follows, we use the following property of H n and V n , whose proofis left to the reader. emma 4.6. Ranging over all objects [ n ] of ∆ , we obtain a cosimplicialstable double category H • . Given a double category D , any double functor H n → D is determined uniquely by a functor [ n ] → H or D , and this identi-fication is compatible with the cosimplicial structure. We obtain bijections Hom DC at ( H n , D ) ∼ = Hom C at ([ n ] , H or D ) which are natural in n , and thus assemble to an isomorphism of simplicialsets Hom DC at ( H • , D ) ∼ = N H or D . The analogous statements hold for V • and V er D . Recall from Definition 1.13 that the path object functor P (cid:67) is definedusing the join construction [ n ] (cid:55)→ [0] ∗ [ n ]. Although [0] ∗ [ n ] ∼ = [ n + 1],it is preferable here to think of adjoining an extra object 0 (cid:48) and using theordering [0] ∗ [ n ] = { (cid:48) ≤ ≤ ≤ . . . ≤ n } . To clarify notation, we write W ([0] ∗ [ n ]) rather than W n +1 for the corre-sponding augmented stable double category.Using this notation, we define a double functor j n : H n → W ([0] ∗ [ n ])by j n ( k ≤ l ) = (0 (cid:48) k (cid:26) (cid:48) l ). For example, for n = 2, we indicate by boldfacethe image of j in the diagram0 (cid:48) (cid:48) (cid:48)
00 01 (cid:48) (cid:48) . Observe that the double functors j n are natural in n , and since 0 (cid:48) remainsunchanged under the cosimplicial structure maps of W ([0] ∗ [ n ]), the functors j n indeed assemble to give a cosimplicial double functor j • .We are now ready to use W • to define the generalized S • -construction. Definition 4.7.
The
Waldhausen S • -construction is the functor S • : DC at aug → s S et which takes an augmented double category D to the simplicial set S • ( D )given by S n ( D ) := Hom DC at aug ( W n , D ) . he first half of our main theorem is given by the following result. Theorem 4.8.
The Waldhausen S • -construction restricts to functors S • : DC at staug → U S eg and S • : DC at st ∗ → U S eg ∗ . We prove this theorem via two propositions. We first need to show that,for any augmented stable double category D , the simplicial set S • ( D ) is 2-Segal. We do so by using the Path Space Criterion (Theorem 1.14), and thefact that a 1-Segal set is just the nerve of a category (Proposition 1.3). Proposition 4.9.
Let D be an augmented stable double category. Thenthere are isomorphisms of simplicial sets P (cid:67) S • ( D ) → N H or D and P (cid:66) S • ( D ) → N V er D . In particular, P (cid:67) S • ( D ) and P (cid:66) S • ( D ) are 1-Segal sets, and hence S • ( D ) isa 2-Segal set.Proof. We establish the first isomorphism; the second one is analogous.If D is a augmented stable double category, then precomposition with thecosimplicial double functor j • gives a map of simplicial sets P (cid:67) S • ( D ) = Hom DC at aug ( W ([0] ∗ • ) , D ) → Hom DC at ( H • , D ) . By Lemma 4.6, it is enough to prove that this map is an isomorphism ofsimplicial sets, which can be accomplished by proving that an augmenteddouble functor F : W ([0] ∗ [ n ]) → D is uniquely determined by its restrictionto the image of j n . To prove this statement for a fixed n , we use inductionon k to show that the row labeled by k ∈ [0] ∗ [ n ] is determined by the imageof j n . For notational purposes, if k = 0, then k − (cid:48) .For the base case k = 0 (cid:48) , observe that if we start with the image of j n only, then to obtain the rest of the 0 (cid:48) row we need only adjoin a horizontalmap to be the image of the map 0 (cid:48) (cid:48) (cid:26) (cid:48)
0. Since D is an augmenteddouble category and we want F to be an augmented functor, the image of0 (cid:48) (cid:48) (cid:26) (cid:48) a (cid:26) F (0 (cid:48)
0) with a inthe augmentation set.Now, let 0 ≤ k < n , and assume that F is uniquely defined up to the rowlabeled by k −
1. To complete the row labeled by k , we first send the map( k − k (cid:16) kk to the unique vertical morphism F (( k − k ) (cid:16) a (cid:48) in D with a (cid:48) ∈ A given by the augmentation.Given k +1 ≤ (cid:96) ≤ n , to define the images of the objects k(cid:96) , the morphisms( k − (cid:96) (cid:16) k(cid:96) and k ( (cid:96) − (cid:26) k(cid:96) , and the square with these morphisms astargets, we proceed by induction on (cid:96) . By stability in D , there is a uniquesquare ,F (( k − (cid:96) − F ( k ( (cid:96) − F (( k − (cid:96) )which we use to extend F .Lastly, if k = n , the image of ( n − n (cid:16) nn is uniquely specified by theaugmentation.We have constructed F on the generating horizontal and vertical mor-phisms, and on the generating squares of W ([0] ∗ [ n ]), which is enough todefine a double functor. Thus, we have established the desired isomorphism.The final statement of the proposition follows from the fact that nerves ofcategories are 1-Segal sets; the fact that S • ( D ) is 2-Segal then follows fromthe Path Space Criterion. (cid:3) Now that we have proved that the image of any augmented stable doublecategory under the S • -construction is a 2-Segal set, it remains to prove thatit is also unital. Proposition 4.10.
Let D be an augmented stable double category. Then S • ( D ) is unital.Proof. By Proposition 4.9 and Lemma 1.11, to prove that S • ( D ) is unital itis enough to check that S ( D ) S ( D ) S ( D ) S ( D ) d s s d and S ( D ) S ( D ) S ( D ) S ( D ) d s s d are pullbacks. We prove that the square on the right is a pullback; theargument for the one on the left is similar.An arbitrary element of the pullback is given by a pair ( a, F ), where a ∈ A = S ( D ) and F is an element of S ( D ), namely, an augmented doublefunctor W → D , which is of the form aa F (12) F (02) F (22) . id ha id va By stability of D , the only square with horizontal source id va is a verticalidentity morphism, so F (02) = F (12) and F is of the form a aa F (12) F (12) F (22) , id ha id va id vF (12) and therefore F is uniquely determined by d F ∈ S ( D ). (cid:3) Now we combine the previous results to show that S • indeed defines thedesired functor. Proof of Theorem 4.8.
Proposition 4.9 implies by Theorem 1.14 that S • ( D )is a 2-Segal set for any augmented stable double category D . Proposi-tion 4.10 shows that S • ( D ) is also unital. Functoriality is immediate bythe definition of S • .Finally, if D is a stable pointed double category, i.e., the augmentationset A consists of exactly one point, then S • D is a reduced unital 2-Segal set,since in this case S D is a single point. (cid:3) The path construction
In this section we construct a functor in the other direction, P : U S eg → DC at staug . We first describe a construction on a unital 2-Segal set, then prove thatits output is an augmented stable double category using Proposition 3.14.Finally, we establish that this assignment is functorial.
Construction 5.1.
Let X be a unital 2-Segal set. By Theorem 1.14, thepath spaces P (cid:67) X and P (cid:66) X are 1-Segal, and hence, by Remark 1.4, we havecategories H = τ P (cid:67) X and V = τ P (cid:66) X , both with X and X as the sets of bjects and morphisms, respectively. The source, target, and identity mapsfor H are given by d , d , and s , respectively, and for V , they are given by d , d , and s . Composition in H and V can be defined by X × X X d ,d ) − −−−−−−→ ∼ = X d −→ X and X × X X d ,d ) − −−−−−−→ ∼ = X d −→ X , respectively.Let Sq = X . We define the horizontal source and the horizontal targetof a square by using the face maps d and d , respectively, as shown in thediagram mor V Sq s h (cid:111) (cid:111) t h (cid:47) (cid:47) mor V X X d (cid:111) (cid:111) d (cid:47) (cid:47) X , and the vertical source and vertical target of a square using the face maps d and d , respectively, as shown in the diagrammor H Sq s v (cid:111) (cid:111) t v (cid:47) (cid:47) mor H X X d (cid:111) (cid:111) d (cid:47) (cid:47) X . More explicitly, for a 3-simplex x ∈ X whose 2-dimensional faces are m, m (cid:48) , e, and e (cid:48) as given in the diagram0 123 e (cid:48) m (cid:48) , em the corresponding square in Sq is given by d d ( x ) d d ( x ) d d ( x ) d d ( x ) d d ( x ) d d ( x ) d d ( x ) d d ( x ) .xm = d ( x ) e = d ( x ) e (cid:48) = d ( x ) m (cid:48) = d ( x ) ote that this particular description of x also establishes that the horizontalsource and target are compatible with the source and target in H , andanalogously for V . Proposition 5.2.
Let X be a unital 2-Segal set. Then the data from Con-struction 5.1 above defines a stable double category P X . Furthermore, theinclusion s X ⊆ X defines an augmentation for P X .Proof. To apply Proposition 3.14 we need to show that:(1) condition (3.12) holds, and(2) the two horizontal “compositions” of squares from (3.13) agree witheach other, and similarly, the two analogous vertical “compositions”of squares coincide.Condition (3.12) follows from 2-Segality of X : every 3-simplex of X iscompletely determined by the pair ( d x, d x ) or by the pair ( d x, d x ), whichin turn are the cospan of targets, or the span of sources,mor V × Ob mor H Sq ( s h ,s v ) (cid:47) (cid:47) ( t h ,t v ) (cid:111) (cid:111) mor V × Ob mor H X × X X X d ,d ) ∼ = (cid:47) (cid:47) ∼ =( d ,d ) (cid:111) (cid:111) X × X X . Thus, once we show that P X is indeed a double category, we know that itis stable.For horizontal composition, consider the following diagram. Starting at X × X X at the left of the diagram and following along the top to X isthe first horizontal composition ◦ h as defined in (3.13); following along the ottom to X is the second horizontal composition ◦ h .(5.3) X X X × X X X × X X ( X × X X ) × X X X × X X X × X ( X × X X ) X × X X X × X X d ( d d , d )( d , d )( d , d d )( d , d ) × d d × ( d , d ) ∼ =id × ( d , d ) id × d ∼ =( d , d )( d , d ) × id ∼ = d × id ( d , d ) ∼ = The 2-Segality of X implies that the map ( d , d ) : X → X × X X is abijection. Indeed, we have the commutative diagram X d ,d ) (cid:47) (cid:47) ( d d ,d d ,d d ) ∼ = (cid:15) (cid:15) X × X X ∼ = ( d ,d ) × ( d ,d ) (cid:15) (cid:15) X × X X × X X ∼ = (cid:47) (cid:47) ( X × X X ) × X ( X × X X )where the two vertical maps are bijections because X is 2-Segal, and thebottom map is an isomorphism by a general limit argument. The left verticalmap is the T -Segal map corresponding to the following triangulation of thepentagon: 0123 4. e have shown that the diagram in (5.3) commutes and thus, that ◦ h = ◦ h = ◦ h : X × X X d ,d ) − −−−−−−→ ∼ = X d −→ X . A similar argument shows that the two vertical compositions coincide andare given by ◦ v = ◦ v = ◦ v : X × X X d ,d ) − −−−−−−→ ∼ = X d −→ X . Lastly, to prove that P X is augmented by s X , note that unitality of X implies that the inner squares are pullbacks in the diagrams Ob (cid:47) (cid:47) (cid:15) (cid:15) A (cid:127) (cid:95) (cid:15) (cid:15) X d (cid:47) (cid:47) s (cid:15) (cid:15) X s (cid:15) (cid:15) s ∼ = (cid:61) (cid:61) X d (cid:47) (cid:47) X mor H s h (cid:47) (cid:47) Ob Ob (cid:47) (cid:47) (cid:15) (cid:15) A (cid:127) (cid:95) (cid:15) (cid:15) X d (cid:47) (cid:47) s (cid:15) (cid:15) X s (cid:15) (cid:15) s ∼ = (cid:61) (cid:61) X d (cid:47) (cid:47) X mor V t v (cid:47) (cid:47) Ob from which it follows that the outer diagrams are pullbacks as well. Thesimplicial identities d s = id and d s = id show that the vertical mapsinteract appropriately with the target in mor H and the source in mor V .The conclusion thus follows from Proposition 3.8. (cid:3) We conclude by proving that the path construction we have defined inthis section is functorial.
Proposition 5.4.
The construction P defines a functor P : U S eg → DC at staug , which restricts to P : U S eg ∗ → DC at st ∗ . Proof.
Let f : X → X (cid:48) be a map between unital 2-Segal sets. For easeof notation, we denote by H or , V er , and Sq the horizontal category ofobjects, the vertical category of objects, and the set of squares of P X , andby H or (cid:48) , V er (cid:48) , and Sq (cid:48) the corresponding ones for P X (cid:48) .Since P (cid:67) and P (cid:66) are functors, we obtain maps of simplicial sets P (cid:67) X → P (cid:67) X (cid:48) and P (cid:66) X → P (cid:66) X (cid:48) , which in turn uniquely determine functors H or f h −→ H or (cid:48) and V er f v −→ V er (cid:48) on the horizontal and vertical categories of objects, respectively. Moreover,both f h and f v agree with f on the set of objects Ob ( P X ) = X . Note hat the assignment is compatible with the augmentation, since f s = s (cid:48) f .The component f : X → X (cid:48) determines a function Sq → Sq (cid:48) between thesets of squares.It remains to check compatibility of the assignment with vertical andhorizontal source and target, identities, and composition. All the argumentsare similar, depending only on the compatibility of f with the simplicialmaps; to illustrate, we establish compatibility with composition using thecommutativity of the following diagram: Sq × V er Sq P f | Sq ×P f | Sq (cid:15) (cid:15) ◦ h (cid:47) (cid:47) Sq P f | Sq (cid:15) (cid:15) X × X X f × f (cid:15) (cid:15) X f (cid:15) (cid:15) ∼ =( d ,d ) (cid:111) (cid:111) d (cid:47) (cid:47) X f (cid:15) (cid:15) X (cid:48) × X (cid:48) X (cid:48) X (cid:48) ∼ =( d (cid:48) ,d (cid:48) ) (cid:111) (cid:111) d (cid:48) (cid:47) (cid:47) X (cid:48) Sq (cid:48) × V er (cid:48) Sq (cid:48) ◦ (cid:48) h (cid:47) (cid:47) Sq (cid:48) . Note that the restriction to reduced unital 2-Segal sets has image inpointed stable double categories and pointed double functors. (cid:3) The equivalence
In this section we prove our main theorem, which states that the functors S • , defined in Definition 4.7, and P , defined in Construction 5.1, are inverseto one another. Theorem 6.1.
The functors S • and P define an equivalence of categories S • : DC at staug (cid:39) ←→ U S eg : P which restricts to an equivalence of categories S • : DC at st ∗ (cid:39) ←→ U S eg ∗ : P . We need to show that there are natural isomorphismsid U S eg ∼ = −→ S • P and P S • ∼ = −→ id DC at staug . The first isomorphism is proven in Proposition 6.2 and the second in Propo-sition 6.3.
Proposition 6.2.
Let X be a unital 2-Segal set. There is a natural isomor-phism of simplicial sets η X : X ∼ = −−→ S • ( P X ) . roof. By Proposition 1.6, since X and S • ( P X ) are 2-Segal sets, it is enoughto define η X at the level of 0-, 1-, 2-, and 3-simplices, and prove that it isan isomorphism at levels 0, 1, and 2.Recall that S ( P X ) is the augmentation of P X , which by construction isthe subset s X of X . Thus, we can define η X at the level of 0-simplices as s : X → S ( P X ) = s X . It is an isomorphism since s is injective.We define η X on 1-simplices as x ∈ X s d x xs d x.s x s x One can check that this output is an allowed 1-simplex in S • ( P X ), and thatthe sources and targets are as stated. Moreover, it is not hard to see thatthis map is injective. The fact that P X is an augmented double category,which follows from the unitality of X , implies surjectivity.We define η X on 2-simplices as x ∈ X s d d x d xs d d x d xd xs d d x.s d xs d x s d xx s d xxs x Again, it is routine to check that this diagram defines an allowed 2-simplexin S • ( P X ). Injectivity is immediate, and surjectivity follows from the factthat P X is stable and augmented.Finally, we define η X on 3-simplices as ∈ X s d d d x d d xs d d d x d d xd d xs d d d x d d xd d xd d xs d d d x.s d d x s d d xd x d xd xs d d xs d d x s d d x d x d xd xs d d xs d x xs d x A standard argument using the simplicial identities shows that these mapsare compatible with the simplicial maps, thus showing that we have con-structed an isomorphism of simplicial sets. (cid:3)
Next, we want to show that the composite P S • : DC at staug → DC at staug isnaturally isomorphic to the identity functor. Proposition 6.3.
Let D be an augmented stable double category. There isa natural isomorphism of double categories ε D : P S • ( D ) ∼ = −−→ D . Proof.
Recall from Remark 1.4 that the fundamental category functor τ is the left adjoint of the nerve functor, with the counit being an isomor-phism. Given an augmented stable double category D , the maps constructedin Proposition 4.9 have adjoints τ P (cid:67) S • ( D ) → H or D and τ P (cid:66) S • ( D ) →V er D which are isomorphisms of categories and are natural in D . By Re-mark 3.15, if there is a double functor ε D , it is already determined by thesetwo isomorphisms. However, it is not immediate that they assemble to adouble functor. Rather than checking their compatibility, we will constructa map on the squares of P S • ( D ) which is compatible with these functors.Recall that the set of squares of P S • ( D ) is given by S ( D ), which in turnis defined to be Hom DC at aug ( W , D ) . To such a double functor F : W → D , let ε D assign the square in D whichis the image of the square F , a construction which is again natural in D . Since both P S • ( D ) and D are stable, to check that we have defined a double functor, it is enoughto check that this assignment is compatible with vertical and horizontalsources and targets. As explained in Construction 5.1, the horizontal sourceand target vertical morphisms are given by d and d , respectively. Recallthat these maps restrict F to the subdiagrams00 0111 120222 and 00 0111 130333and the identification with V er restricts these two diagrams to the maps02 (cid:16)
12 and 03 (cid:16)
13, respectively. But these maps are exactly the hori-zontal source and target vertical morphisms of the square indicated abovein D . Similarly, the described map is also compatible with vertical sourceand target.Next, we have to check that the double functor ε D : P S • ( D ) → D isaugmented. Recall that the augmentation of P S • ( D ) is given by the imageof the degeneracy map s : S ( D ) → S ( D ). An augmented double functor F : W → D is in the image of s if and only if it is of the form F (00) F (01) F (11) . In particular, since F is augmented, the object F (01) of D to which thisobject of P S • ( D ) is sent is an object in the augmentation set of D . Thusthe double functor P S • ( D ) → D is augmented.We already know that ε D induces isomorphisms on H or and V er , so itremains to show that it induces a bijection on the set of squares. Given anysquare xz y in D , we have to construct an augmented double functor G : W → D whichmaps 130212 03to it. Define G (11 (cid:26)
12) and G (12 (cid:16)
22) to be the unique elements a (cid:26) z and z (cid:16) a in Hor D and V er D , respectively, with a , a in theaugmentation set A . By stability of G , the following span and cospan canbe completed uniquely into squares, as indicated in the diagrams zx a x and x .za w We set these squares to be the images, respectively, of the squares120111 02 and 231222 13under G . Finally, we use the augmentation again to produce unique maps a (cid:26) x and x (cid:16) a which are declared to be the images of 00 (cid:26) (cid:16)
33 under G , respectively. Thus we have defined the desireddouble functor G : W → D . Note that there were no choices involved in theconstruction of G , so G is indeed the unique preimage of the given squareunder the map of squares of the double functor ε D : P S • D → D . (cid:3) Three examples, revisited
In this section, we return to the examples of 2-Segal sets described inSection 2 and construct their corresponding augmented stable double cate-gories.
Example . Recall the 2-Segal set M • which is the nerve of a partial monoidfrom Example 2.1. Let us consider the image of this 2-Segal set under thepath construction. he objects of the associated double category P M are the elements ofthe monoid a ∈ M . The set of horizontal morphisms, which is equal tothe set of vertical morphisms, is the set of composable pairs M . However,their interpretation is different: for ( a, b ) ∈ M , the corresponding horizontalarrow has source a and target a · b , i.e., it can be interpreted as multiplicationon the right by b , a a · b. · b The vertical arrow corresponding to ( a, b ) ∈ M has target b and source a · b ,so it can be thought of as multiplication on the left by a , but with the arrowpointing in the other direction. a · bb. a · The set of squares is M and reflects associativity of the multiplication: for( a, b, c ) ∈ M , its associated square is a · b ( a · b ) · c = a · ( b · c ) b b · c. a · · c a ·· c The double category P M is pointed by the unit 1 ∈ M , since for every a ∈ M , the elements (1 , a ) ∈ M and ( a, ∈ M , which can be visualizedas 1 1 · a = a · a and a = a · , a · exhibit 1 as initial with respect to the horizontal category and terminalwith respect to the vertical category. Finally, stability is again given byassociativity, since both a · b ( a · b ) · cb a · · c and a · ( b · c ) b b · c a ·· c can be completed uniquely to a square as above. Example . Let us now revisit the 2-Segal set 2Cob ≤ g from Example 2.2.The objects of the associated double category are elements in (2Cob ≤ g ) ,which are diffeomorphism classes of 2-dimensional cobordisms Σ with genusat most g . Horizontal and vertical morphisms are elements in (2Cob ≤ g ) ,which are given by diffeomorphism classes of cobordisms Σ with genus atmost g , together with a choice of decomposition Σ ∼ = Σ (cid:113) N Σ , where = ∂ out Σ = ∂ in Σ . We view Σ as a horizontal morphism and as a verticalmorphism viaΣ Σ (cid:113) N Σ ∼ = Σ ( − ) (cid:113) N Σ and Σ ∼ = Σ (cid:113) N Σ Σ , Σ (cid:113) N ( − ) respectively.The augmentation is given by cylinders on 1-dimensional closed manifoldsviewed as trivial cobordisms, since the set of such cylinders is the image of(2Cob ≤ g ) under the degeneracy map s .Given an object in the double category Σ ∈ (2Cob ≤ g ) , there is a uniqueobject in the augmentation, namely the cylinder on its incoming boundary,together with a unique horizontal morphism to Σ: ∂ in Σ × [0 ,
1] ( ∂ in Σ × [0 , (cid:113) ∂ in Σ Σ ∼ = Σ . ( − ) (cid:113) ∂ inΣ Σ For example, for the pair of pants, we have the horizontal morphism ∼ = . q ( − ) Similarly, there is a unique object in the augmentation, namely the cylinderon its outgoing boundary, together with a unique vertical morphism fromΣ: Σ ∼ = Σ (cid:113) ∂ out Σ ( ∂ out Σ × [0 , ∂ out Σ × [0 , . Σ (cid:113) ∂ outΣ ( − ) In the example of the pair of pants, the vertical morphism is (drawn hori-zontally) ∼ = . q ( − ) Example . Finally, we apply our construction to the 2-Segal set X asso-ciated to a graph G as described in Example 2.3. n element in X , such as a b represents a horizontal morphism a a b a b. The same element gives the vertical morphism a bb.a b
To give an idea of the different way squares can be formed from suchmorphisms, we give a few examples. A 3-simplex such as a b c gives rise to a square b a b ca b cba b b c.a b cb c However, squares can look fairly different even if we permute the sets inthe partition. For example, the 3-simplex a b c b c a b cc a cb c a b ca b ca c
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Department of Mathematics, University of Virginia, Charlottesville, VA22904, USA
E-mail address : [email protected] Department of Mathematics, Reed College, Portland, OR 97202, USA
E-mail address : [email protected] Mathematical Institute, University of Bonn, 53115 Bonn, Germany
E-mail address : [email protected] UPHESS BMI FSV, ´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015Lausanne, Switzerland
E-mail address : [email protected] Max Planck Institute for Mathematics, 53111 Bonn, Germany
E-mail address : [email protected]@mpim-bonn.mpg.de