A cobordism model for Waldhausen K -theory
aa r X i v : . [ m a t h . K T ] O c t A COBORDISM MODEL FOR WALDHAUSEN K -THEORY GEORGE RAPTIS AND WOLFGANG STEIMLE
Abstract.
We study a categorical construction called the cobordism category ,which associates to each Waldhausen category a simplicial category of cospans.We prove that this construction is homotopy equivalent to Waldhausen’s S • -construction and therefore it defines a model for Waldhausen K -theory. As anexample, we discuss this model for A -theory and show that the cobordism cat-egory of homotopy finite spaces has the homotopy type of Waldhausen’s A ( ∗ ).We also review the canonical map from the cobordism category of manifoldsto A -theory from this viewpoint. Introduction
Many of the definitions of higher algebraic K -theory are fundamentally based oncategorical constructions. Some of the main categorical constructions are Quillen’soriginal Q -construction for exact categories [6], the closely related s • -construction,Waldhausen’s more general S • -construction for Waldhausen categories [12], andThomason’s T • -construction. To this incomplete list, we may also add the Gillet-Grayson G -construction and Quillen’s S − S -construction. Moreover, ∞ -categoricalversions of such constructions have also been developed in recent years (see, forexample, [1]). Where it applies, each of these constructions leads to the same (=homotopy equivalent) definition of higher algebraic K -theory.In this paper we introduce a new such construction for Waldhausen categories.The construction applies, in fact, more generally to unpointed Waldhausen cate-gories where neither an initial nor a terminal object is required. This constructionis obtained from categories of cospans where one of the arrows is a cofibration. Wecall this construction the cobordism category partly because our inspiration for thisconstruction came from previous work [7, 8] on the relation between the cobordismcategory of manifolds [4, 5] with the algebraic K -theory of spaces ( A -theory) [12].Moreover, specifically in this context, the idea to regard a cospan of homotopyfinite spaces as a kind of formal cobordism seems particularly illuminating. Con-versely, this connection also inspired in [11] the use of K -theoretic methods in thestudy of cobordism categories of manifolds. The main result of the present papercompares this cobordism category construction with the S • -construction and showsthat the loop space of the classifying space of this cobordism category is homotopyequivalent to Waldhausen K -theory.The cobordism category construction is reminiscent of the Q -construction [6, 1].Each one is related to the S • -construction via Segal’s edgewise subdivision [10],but in “opposite” ways. The cobordism category can be compared with the Q -construction directly by taking (homotopy) pullbacks/pushouts in order to ex-change a sequence of cospans with a sequence of spans. However, our proof thatthe cobordism category models Waldhausen K -theory does not require a biWald-hausen category structure and it applies to all Waldhausen categories. Thus, one could consider our results as a proof that a variation of the Q -construction yieldsthe correct homotopy type even in cases which are not stable or additive. In futurework, we plan to use the cobordism category model in the comparison betweencobordism categories of manifolds and the algebraic K -theory of spaces, followingthe work initiated in [9].In this direction, we also discuss in the present paper models for A -theory basedon the cobordism category construction. More specifically, we prove in Theorem 4.4that the algebraic K -theory A ( X ) of a space X can be obtained from the classifyingspace of a category of formal cobordisms between homotopy finite spaces with astructure map to X . Using this cobordism model for A ( X ), we can describe themap from the standard cobordism category of manifolds to Waldhausen’s A -theoryessentially as an inclusion of cobordism categories.The paper is organized as follows. In Section 2, we define the cobordism cate-gory of an unpointed Waldhausen category and discuss some of its properties. Asthe construction uses Segal’s edgewise subdivision, we begin with a short reviewof this subdivision. In a final subsection, we discuss a variant of the cobordismcategory where each cospan consists of “disjoint” cofibrations. In Section 3, wedefine a comparison map from the cobordism category to the S • -construction. Ourmain result (Theorem 3.1) shows that this induces a homotopy equivalence aftergeometric realization. We also explain that the cobordism category constructioncan be iterated so that it can also be used to obtain the deloopings of Waldhausen K -theory. In Section 4, we discuss cobordism category models for A -theory andgive a definition of the map from the standard cobordism category of manifolds toWaldhausen’s A -theory using this model. Acknowledgements.
We warmly acknowledge the support and the hospitality of theHausdorff Institute for Mathematics in Bonn where a preliminary outline of thiswork was completed. The first named author was partially supported by
SFB 1085— Higher Invariants (University of Regensburg) funded by the DFG. The secondnamed author was partially supported by the DFG priority programme
SPP 2026— Geometry at infinity .2.
The cobordism category of a Waldhausen category
Edgewise subdivision.
We recall Segal’s edgewise sudivision Sd( X ) of asimplicial set X (see also [10, Appendix 1]). Let ∆ denote the usual category of finiteordinals [ n ] = { < < · · · < n } and order-preserving maps. Let ( − ) op : ∆ → ∆be the standard involutive functor with [ n ] op = [ n ] and α op ( k ) = n − α ( m − k )for each α : [ m ] → [ n ] in ∆. Then we define a functor µ : ∆ ( − ) op × id −−−−−−→ ∆ × ∆ −∗− −−−→ ∆where ∗ denotes the ordinal sum. It will be convenient to represent the well-orderedfinite set µ [ n ] = [ n ] op ∗ [ n ] = [2 n + 1] as follows: { n < n − < · · · < < < · · · < n } . Given a simplicial set X : ∆ op → Set, the edgewise subdivision is defined bySd( X ) : ∆ op µ op −−→ ∆ op X −→ Set . COBORDISM MODEL FOR WALDHAUSEN K -THEORY 3 More specifically, we have that (Sd X ) n = X n +1 and the simplicial operators ofSd( X ) are determined by X using the presentation of [2 n + 1] shown above. Clearlythis defines an endofunctor on the category of simplicial sets,Sd : SSet → SSet , which is induced by the endofunctor µ .When X is the nerve of a small category C , then Sd( X ) is also the nerve of acategory, namely, the twisted arrow category tw C of C . Its objects are the arrows c → d in C , and a morphism from c → d to c ′ → d ′ is a commutative square in C c / / d (cid:15) (cid:15) c ′ O O / / d ′ . The composition in tw C is given by concatenating such squares vertically, thencomposing vertical arrows and forgetting the intermediate horizontal arrow.There is a canonical homeomorphism | X | ∼ = | Sd( X ) | which, however, does notarise from a simplicial map, and will not be used in this paper. We will instead usea natural comparison map between simplicial sets called the last-vertex map L : Sd( X ) → X which is induced by the canonical inclusions [ n ] ⊂ [ n ] op ∗ [ n ] for each [ n ] ∈ ∆. Inthe case of (nerves of) categories, the last-vertex map corresponds to the functortw C → C which sends ( c → d ) to d and ( c → d ) → ( c ′ → d ′ ) to d → d ′ .The last-vertex map L : Sd( X ) → X is known to be a natural weak equivalenceof simplicial sets. (A proof of this claim can be given using the following standardmethod. Let E denote the class of simplicial sets X such that L : Sd( X ) → X is a weak equivalence. We first note that ∆ n , n ≥
0, is in E , for Sd(∆ n ) is thenerve of the category tw[ n ] and this has a terminal object. Moreover, the functorSd commutes with colimits and it preserves monomorphisms. It follows that E is closed under pushouts along a monomorphism and under directed colimits ofsimplicial sets. Lastly, using the skeletal filtration of a simplicial set, we concludethat the class E contains all simplicial sets.)2.2. Definition of the cobordism category.
For the definition and the basicproperties of the cobordism category construction, it will suffice to work with anunpointed version of the notion of a Waldhausen category [12]. By an unpointedWaldhausen category we mean a small category C equipped with subcategories ofcofibrations co C ⊆ C and of weak equivalences w C ⊆ C that satisfy the followingaxioms (compare [12]):(1) Both co C and w C contain the isomorphisms in C .(2) For any morphism C → X and any cofibration C D , the pushout X ∪ C D exists in C , and the induced map X → X ∪ C D is again a cofibration.(3) The weak equivalences satisfy the glueing lemma [12, p. 326].This list contains exactly those of the axioms for a Waldhausen category that do notinvolve a zero object. An exact functor between unpointed Waldhausen categories F : C → C ′ is a functor which preserves cofibrations, weak equivalences, and thosepushout squares of the form described in axiom (2) above. GEORGE RAPTIS AND WOLFGANG STEIMLE
Clearly every Waldhausen category is an unpoined Waldhausen category. On theother hand, the categories of (unpointed) finite sets, and of (unbased) homotopyfinite spaces are not pointed and hence do not underlie a Waldhausen category, butdo admit the structure of an unpointed Waldhausen category. The latter example,and generalizations thereof, will be discussed in Section 4.1.Let C be an unpointed Waldhausen category and let Cat denote the category ofsmall categories. We define a simplicial category Cob ( C ) • : ∆ op → Cat as follows.The category
Cob ( C ) n is the category of functors F : tw[ n ] → C , ( i ≤ j ) F ij such that(i) for each i ≤ j ≤ k , the map F ij → F ik is a cofibration, and(ii) for each i ≤ j ≤ k ≤ l , the diagram in C F il F ik = = = = ④④④④④④④④ F jl a a ❈❈❈❈❈❈❈❈ F jk = = = = ⑤⑤⑤⑤⑤⑤⑤⑤ a a ❈❈❈❈❈❈❈❈ is a pushout square.The morphisms in Cob ( C ) n are the natural transformations between such functors.We denote w Cob ( C ) n ⊂ Cob ( C ) n the subcategory of objectwise weak equivalences.Thus, Cob ( C ) = C and w Cob ( C ) = w C . The category Cob ( C ) (resp., w Cob ( C ) )consists of diagrams in C of the form F F = = = = ③③③③③③③③ F a a ❉❉❉❉❉❉❉❉ and natural transformations (resp., natural weak equivalences) between them. Wefind it useful to think of such a diagram as some kind of (formal) cobordism from F to F . In the next simplicial degree an object in Cob ( C ) can be depicted asfollows: F F = = = = ③③③③③③③③ F a a ❉❉❉❉❉❉❉❉ F = = = = ③③③③③③③③ F = = = = ③③③③③③③③ a a ❉❉❉❉❉❉❉❉ F a a ❉❉❉❉❉❉❉❉ where the middle square is a pushout, and similarly for the higher simplicial de-grees. In the language of cobordisms, the last diagram corresponds to the datumof two composable cobordisms together with a choice of a composition. Continuingthis analogy, an object in Cob ( C ) n corresponds to a string of n many composablecobordisms, together with choices of compositions for all connected substrings. COBORDISM MODEL FOR WALDHAUSEN K -THEORY 5 Both
Cob ( C ) n and w Cob ( C ) n are natural in [ n ] and define simplicial objects inthe category of small categories Cat. Moreover, Cob ( C ) • also defines a simplicialobject in the category of unpointed Waldhausen categories and exact functors. (Wewill return to this point in Subsection 3.3.) The simplicial category Cob ( C ) • (resp. w Cob ( C ) • ) is a type of cospan category with restrictions, as imposed by condition(i). Definition 2.1.
The cobordism category of the unpointed Waldhausen category C is the simplicial space Cob ( C , w C ) : [ n ] (cid:12)(cid:12) N • w Cob ( C ) n (cid:12)(cid:12) . The geometric realization of this simplicial space is called the classifying space ofthe cobordism category and will be denoted by
B Cob ( C , w C ).The definition of the cobordism category is clearly functorial with respect toexact functors between (unpointed) Waldhausen categories. Remark . It is easy to see that w Cob ( C ) defines a Segal object in Cat, in thesense that the canonical restriction along the spine inclusion w Cob ( C ) n ≃ −→ w Cob ( C ) × w Cob ( C ) · · · × w Cob ( C ) w Cob ( C ) | {z } n is an equivalence of categories for each n ≥
1. Since geometric realization commuteswith pullbacks, the cobordism category is also a Segal object in (a convenient modelfor) the category of spaces. This provides some justification of the term cobordism category . However, we do not know at this level of generality whether
Cob ( C , w C )is a Segal space in the usual sense – that is, if additionally the iterated pullbacks | w Cob ( C ) | × | w Cob ( C ) | · · · × | w Cob ( C ) | | w Cob ( C ) | are also homotopy pullbacks. (This is true, for instance, if w C is the class ofisomorphisms in C .)2.3. Relative isomorphisms.
Our main goal is to establish an equivalence be-tween the cobordism category construction and Waldhausen’s S • -construction inthe case of a standard Waldhausen category. This will be done in Section 3. Themain ingredient in the proof of this equivalence is the following useful method forshowing that certain functors between cobordism categories induce homotopic mapsbetween the classifying spaces.Let C be an unpointed Waldhausen category. We denote by Cob po ( C ) • ⊂ Cob ( C ) [1] • the simplicial subcategory which is given in degree n ≥ Cob ( C ) [1] n that is spanned by the functors F : tw[ n ] × [1] → C , ( i ≤ j, a ) F aij such that, for each i ≤ j ≤ k , the diagram in C F ij / / / / (cid:15) (cid:15) F ik (cid:15) (cid:15) F ij / / / / F ik GEORGE RAPTIS AND WOLFGANG STEIMLE is a pushout square. We also denote by w Cob po ( C ) • ⊂ Cob po ( C ) • the subcate-gory of objectwise weak equivalences. Note that there are morphisms of simplicialcategories, Cob ( C ) • δ • / / Cob po ( C ) • s • / / t • / / Cob ( C ) • which are induced by the obvious maps [0] ⇒ [1] → [0]. These morphisms alsorestrict to the simplicial subcategories of weak equivalences. Lemma 2.3.
Let C be an unpointed Waldhausen category. (a) The inclusion of simplicial sets δ • : ob( Cob ( C ) • ) → ob( Cob po ( C ) • ) is a weak equivalence (i.e., it induces a weak homotopy equivalence aftergeometric realization). (b) The inclusion of simplicial sets δ • : N n w Cob ( C ) • → N n w Cob po ( C ) • is a weak equivalence for each n ≥ . As a consequence, there is a homotopyequivalence δ : B Cob ( C , w C ) ≃ −→ (cid:12)(cid:12) N • w Cob po ( C ) • (cid:12)(cid:12) . (c) The source and target projections s • and t • induce homotopic maps s ≃ t : (cid:12)(cid:12) N • w Cob po ( C ) • (cid:12)(cid:12) → B Cob ( C , w C ) . Proof.
Note that s • ◦ δ • = id = t • ◦ δ • . Thus, for (a), it is enough to specify asimplicial homotopy H : ob( Cob po ( C ) • ) × ∆ → ob( Cob po ( C ) • )between δ • ◦ s • and the identity map. This homotopy sends ( F, α : [ n ] → [1]) to thecomposite tw[ n ] × [1] (id ,q ) × id −−−−−−→ tw[ n ] × [ n ] op × [1] id × α op × id −−−−−−−−→ tw[ n ] × [1] op × [1] id × h −−−−→ tw[ n ] × [1] F −→ C where the first map is induced by the first-vertex map q : tw[ n ] → [ n ] op , given by( i ≤ j ) i , and the map h : [1] op × [1] → [1] is the standard (reverse) homotopybetween id and the constant map at 0, i.e., h (1 , i ) = 0 and h (0 , i ) = i .It follows from the definition that this is a simplicial map but we need to checkthat it really lands in Cob po ( C ) • . Let G denote the image of ( F, α ) under H .Then for fixed α , we have that G ij = F ij and G ij is either F ij or F ij , dependingon whether α ( i ) is 0 or 1, respectively. From this it follows easily that for each i ≤ j ≤ k and any a = 0 ,
1, the map G aij G aik is a cofibration. Moreover, using our assumptions on F , it follows that for each i ≤ j ≤ k ≤ l and any a = 0 ,
1, the diagram G ajk / / / / (cid:15) (cid:15) G ajl (cid:15) (cid:15) G aik / / / / G ail COBORDISM MODEL FOR WALDHAUSEN K -THEORY 7 is a pushout square. Similarly, for each i ≤ j ≤ k , the diagram G ij / / / / (cid:15) (cid:15) G ik (cid:15) (cid:15) G ij / / / / G ik is also a pushout square. Therefore the simplicial homotopy is well-defined and (a)follows.(b) We apply (a) to the unpointed Waldhausen category w n C whose objects are n -strings of weak equivalences in C , and whose morphisms are natural transformationsof diagrams. This is a full subcategory of C [ n ] and is regarded as an unpointedWaldhausen category with the objectwise structure. Then the claim follows sinceob( Cob ( w n C ) • ) ∼ = N n w Cob ( C ) • , ob( Cob po ( w n C ) • ) ∼ = N n w Cob po ( C ) • . (c) is an immediate consequence of (the second part of) (b). (cid:3) Remark . It is also possible to establish an additivity theorem for the cobor-dism category, analogous to Waldhausen’s Additivity Theorem, by using similararguments and following Waldhausen’s proof in [12].
Definition 2.5.
Let
F, G : C → D be exact functors between unpointed Wald-hausen categories. A natural transformation φ : F → G is called a relative isomor-phism if for each cofibration f : c c ′ in C , the diagram in D F ( c ) / / F ( f ) / / φ c (cid:15) (cid:15) F ( c ′ ) φ c ′ (cid:15) (cid:15) G ( c ) / / G ( f ) / / G ( c ′ )is a pushout square.As an immediate consequence of Lemma 2.3, we have the following proposition. Proposition 2.6.
Let
F, G : C → D be exact functors between unpointed Wald-hausen categories and let φ : F → G be a relative isomorphism. Then the inducedmaps B Cob ( F, wF ) ≃ B Cob ( G, wG ) :
B Cob ( C , w C ) → B Cob ( D , w D ) are homotopic.Proof. The natural transformation φ : C → D [1] defines an exact functor of un-pointed Waldhausen categories, where D [1] is equipped with the objectwise struc-ture. Then we obtain a simplicial functorΦ • : = Cob ( φ ) • : Cob ( C ) • → Cob ( D [1] ) • = Cob ( D ) [1] • , which by assumption lands in the simplicial subcategory Cob po ( D ) • . Moreover,Φ • : Cob ( C ) • → Cob po ( D ) • is such that s • ◦ Φ • and t • ◦ Φ • correspond to thesimplicial functors induced by F and G respectively. Then the result follows fromLemma 2.3(c). (cid:3) GEORGE RAPTIS AND WOLFGANG STEIMLE
Example . Let C be an unpointed Waldhausen category which has an initialobject ∅ such that the unique morphism ∅ → X is a cofibration for every object X in C (see also Section 2.4). Then for any object X in C , there is an exact “shift”functor − ∐ X : C → C . The canonical natural transformation from the identityfunctor to this shift functor ( − ∐ X ) is a relative isomorphism. By Proposition 2.6,it follows that the map induced on the cobordism category by the shift functor ishomotopic to the identity.2.4. A symmetric version.
The definition of the cobordism category construc-tion is not symmetric under passing to the opposite category, because of the cofibra-tion condition. In this subsection we present a symmetric variant of the cobordismcategory and show that it agrees with the original one in most cases of interest.Let C be an unpointed Waldhausen category which has an initial object ∅ suchthat the unique morphism ∅ → X is a cofibration for each object X ∈ C (so that, inparticular, C has finite coproducts). We will refer to such an unpointed Waldhausencategory as an unpointed Waldhausen category with initial object .For each n ≥
0, we consider the full subcategory
Cob sym ( C ) n ⊂ Cob ( C ) n which is spanned by the functors F : tw[ n ] → C in Cob ( C ) n such that in addition(iii) for each 0 ≤ i < n , the canonical map F ii ⊔ F i +1 ,i +1 → F i,i +1 is a cofibration.Under this extra assumption, for each 0 ≤ i ≤ j ≤ k ≤ n , the map F jk → F ik isa cofibration, too. Then Cob sym ( C ) • : [ n ] Cob sym ( C ) n defines a semi-simplicialobject in Cat, i.e., a functor on the subcategory ∆ op < ⊂ ∆ op which consists ofthe injective maps. (In the category–theoretic interpretation of Cob sym ( C ) • , theabsence of degeneracies can be interpreted as the absence of identity morphisms.)We also have the corresponding semi-simplicial subcategory of weak equivalences w Cob sym ( C ) • ⊂ Cob sym ( C ) • . Remark . The construction
C 7→
Cob sym ( C ) • works also under the weaker as-sumption on C , namely, that a pushout of D ← C → X exists in C when both maps C → D and C → X are cofibrations. Definition 2.9.
Let C be an unpointed Waldhausen category with initial object.The symmetric cobordism category of C is the semi-simplicial space Cob sym ( C , w C ) : [ n ] (cid:12)(cid:12) N • w Cob sym ( C ) n (cid:12)(cid:12) . The geometric realization of this semi-simplicial space is called the classifying space of the symmetric cobordism category and will be denoted by
B Cob sym ( C , w C ).We compare this symmetric construction with the previous cobordism categoryconstruction in the case where C has functorial factorizations. We recall that C is said to have functorial factorizations if every morphism in C can be writtenfunctorially in the arrow category C [1] as the composition of a cofibration followed bya weak equivalence. For Waldhausen categories, this is equivalent to the existenceof a cylinder functor satisfying the cylinder axiom in the sense of [12, 1.6].The inclusion (of semi-simplicial categories) w Cob sym ( C ) • ⊂ w Cob ( C ) • induces amap between the geometric realizations of the corresponding semi-simplicial spaces. COBORDISM MODEL FOR WALDHAUSEN K -THEORY 9 This can be composed with the canonical homotopy equivalence to the classifyingspace of
Cob ( C , w C ) • , so we obtain a canonical map B Cob sym ( C , w C ) → B Cob ( C , w C ) . Proposition 2.10.
Let C be an unpointed Waldhausen category with initial object.Suppose that C has functorial factorizations. Then the inclusion map B Cob sym ( C , w C ) → B Cob ( C , w C ) is a weak homotopy equivalence.Proof. It suffices to prove that the inclusion of semi-simplicial categories induces ahomotopy equivalence in each simplicial degree of the cobordism direction. Usingthe functorial factorizations, we can functorially replace each cospan F F = = = = ③③③③③③③③ F a a ❉❉❉❉❉❉❉❉ by a weakly equivalent cospan F F = = = = ④④④④④④④④ F a a a a ❈❈❈❈❈❈❈❈ where F ⊔ F F ∼ −→ F is the functorial factorization. Repeating thisprocess and making choices of pushouts, we obtain a functor r : w Cob ( C ) n → w Cob sym ( C ) n such that both composites w Cob ( C ) n r −→ w Cob sym ( C ) n incl ֒ → w Cob ( C ) n w Cob sym ( C ) n incl ֒ → w Cob ( C ) n r −→ w Cob sym ( C ) n are naturally weakly equivalent to the respective identity functors. Thus, the in-clusion map is a homotopy equivalence for each n ≥
0, and the result follows. (cid:3)
Remark . There is an intermediate object between
Cob sym ( C ) • and Cob ( C ) • where instead of (iii) above, we require only that the maps F i +1 ,i +1 → F i,i +1 arecofibrations. This defines a simplicial subobject of Cob ( C ) • and the associatedclassifying space has again the same homotopy type when C has functorial factor-izations. 3. Comparison with the S • -construction Recollections.
We recall Waldhausen’s S • -construction from [12]. Let C bea (pointed) Waldhausen category and let Ar[ n ] : = [ n ] [1] denote the arrow categoryof [ n ]. The notation for an object ( i ≤ j ) of Ar[ n ] will be often abbreviated to ( ij ).The category S n C ⊂ C
Ar[ n ] is the full subcategory spanned by the functors F : Ar[ n ] → C , ( i ≤ j ) F ij such that (i) for each i , F ii = ∗ ,(ii) for each i ≤ j ≤ k , the map F ij → F ik is a cofibration,(iii) for each i ≤ j ≤ k ≤ l , the diagram in C F ik / / / / (cid:15) (cid:15) F il (cid:15) (cid:15) F jk / / / / F jl is a pushout square.The morphisms in S n C are given by natural transformations. The category S n C car-ries a natural Waldhausen category structure (see [12]) where the weak equivalencesare the objectwise weak equivalences between functors. We denote by wS n C ⊂ S n C the subcategory of weak equivalences in S n C . Both S n C and wS n C are natural in [ n ]and they define simplicial objects in Cat. Moreover, S • C is a simplicial object in thecategory of Waldhausen categories and exact functors of Waldhausen categories.By definition, we have S C = {∗} and S C = C . S C is the category of cofibersequences in C . In the next simplicial degree, an object in S C can be depicted asa triangular staircase diagram of the form ∗ / / / / F / / / / (cid:15) (cid:15) F / / / / (cid:15) (cid:15) F (cid:15) (cid:15) ∗ / / / / F / / / / (cid:15) (cid:15) F (cid:15) (cid:15) ∗ / / / / F (cid:15) (cid:15) ∗ where each embedded square is a pushout. Following [12], we denote by F n C ⊂ C [ n ] the full subcategory spanned by filtered objects, i.e., the functors F : [ n ] → C , i F i such that F i → F j is a cofibration for each i ≤ j . Then the restriction functor S n C → F n − C , ( F ij ) ≤ i ≤ j ≤ n ( F j +1 ) ≤ j ≤ n − is an equivalence of categories because S n C is obtained from F n − C simply bymaking choices of pushouts for each filtered object - however, F •− C do not definea simplicial object.The algebraic K -theory K ( C ) of C [12] is defined to be the loop space (basedat the point represented by the zero object) of the geometric realization of thesimplicial space [ n ] (cid:12)(cid:12) N • wS n C (cid:12)(cid:12) , i.e., K ( C ) : = Ω (cid:12)(cid:12) N • wS • C (cid:12)(cid:12) . The comparison map.
Let C be a Waldhausen category. We will comparethe cobordism category of C with the S • -construction of C . The comparison map COBORDISM MODEL FOR WALDHAUSEN K -THEORY 11 is essentially defined by functors Cob ( C ) n → F n − ( C ) , F F F · · · F n F ! . In order to promote this collection of functors to a simplicial functor mapping intothe S • -construction, we must first modify our model for Cob ( C ) • so that it includeschoices of pushouts. For that purpose, we consider the full subcategory f Ar[ n ] ⊂ Ar[ n ]spanned by objects ( i ≤ j ) where ¯ i ≤ j ; here ( − ) denotes the order-reversing self-isomorphism of [ n ] given by i n − i . Thus, for example, the object (1 ≤
2) ofAr[3] is an object of f Ar[3] because ¯1 = 2 ≤
2. In more detail, the subposet f Ar[3] isexactly the lower right half of the depicted Ar[3](00) / / (01) / / (cid:15) (cid:15) (02) / / (cid:15) (cid:15) (03) (cid:15) (cid:15) ⑤⑤⑤⑤ (11) / / (12) / / ③③③③③ ③ ③ ③ ③ (cid:15) (cid:15) (13) (cid:15) (cid:15) (22) / / (23) (cid:15) (cid:15) (33)with respect to the indicated line that passes through (03). Similarly, f Ar[ n ] ⊂ Ar[ n ]corresponds to its “lower right half” with respect to the dichotomizing line thatpasses through (0 ≤ n ).More specifically, we will consider the category f Ar[2 n + 1] = f Ar([ n ] op ∗ [ n ]) (usingthe canonical identification of [2 n + 1] with [ n ] op ∗ [ n ]). An object of f Ar[2 n + 1] iseither of the form ( i ≤ j ), for any n < i ≤ j ≤ n +1, or of the form ( n − i ≤ j ) where0 ≤ i ≤ n < j ≤ n . The first collection of objects defines a subposet isomorphic toAr[ n ], while the second collection defines a subposet isomorphic to tw[ n ]. Thus, wehave full inclusions Ar[ n ] ⊂ f Ar([ n ] op ∗ [ n ]) ⊃ tw[ n ] . Here Ar[ n ] includes as objects of the first kind which form a smaller triangularstaircase at the lower right part of the staircase diagram. The second inclusionof tw[ n ] includes the objects of the second kind. These subposets have emptyintersection in f Ar([ n ] op ∗ [ n ]). In the example of f Ar[3] above, the inclusion Ar[1] ⊂ f Ar[3] corresponds to the subposet(22) / / (23) (cid:15) (cid:15) (33) while the inclusion tw[1] ⊂ f Ar[3] corresponds to the subposet(13)(03) < < ③③③③③③③③ (12) . b b ❊❊❊❊❊❊❊❊ We define the modified cobordism category
Cob big ( C ) • to be the simplicial cate-gory which in degree n ≥ F : f Ar([ n ] op ∗ [ n ]) → C satisfying the conditions in the S • -construction, namely:(i) for each i , F ii = ∗ ,(ii) for each i ≤ j ≤ k , the map F ij → F ik is a cofibration,(iii) for each i ≤ j ≤ k ≤ l , the diagram in C F ik / / / / (cid:15) (cid:15) F il (cid:15) (cid:15) F jk / / / / F jl is a pushout square.The morphisms in this category are given by natural transformations between suchfunctors. We denote by w Cob big ( C ) n ⊂ Cob big ( C ) n the subcategory of objectwiseweak equivalences. Note that each F ∈ Cob big ( C ) n is determined up to canonicalisomorphism by its restriction alongtw[ n ] ⊂ f Ar([ n ] op ∗ [ n ])since the rest of the diagram can be obtained by making choices of pushouts in C .( F ij is the cofiber of F ¯0 i F ¯0 j .) Thus, the restriction functors Cob big ( C ) • → Cob ( C ) • , w Cob big ( C ) • → w Cob ( C ) • are degreewise equivalences of categories, and therefore they induce homotopyequivalences between the geometric realizations. In particular, we have a canonicalhomotopy equivalence(1) (cid:12)(cid:12) N • w Cob big ( C ) • (cid:12)(cid:12) ≃ B Cob ( C , w ) . Then we define a morphism between simplicial objects in Cat τ • : w Cob big ( C ) • → wS • C induced by the restriction along the inclusion Ar[ n ] ⊂ f Ar([ n ] op ∗ [ n ]). Passing tothe geometric realizations, we obtain a natural (zigzag) comparison map of spaces τ : B Cob ( C , w C ) ≃ (cid:12)(cid:12) N • w Cob big ( C ) • (cid:12)(cid:12) −→ (cid:12)(cid:12) N • wS • C (cid:12)(cid:12) . Theorem 3.1.
The comparison map τ is a weak homotopy equivalence. COBORDISM MODEL FOR WALDHAUSEN K -THEORY 13 Proof.
We may apply Segal’s edgewise subdivision to the simplicial object wS • C and obtain a new simplicial category Sd wS • C with (Sd wS • C ) n = wS n +1 C . Therestriction along the inclusion f Ar([ n ] op ∗ [ n ]) ⊂ Ar([ n ] op ∗ [ n ]) induces a simplicialfunctor α • : Sd wS • C → w Cob big ( C ) • such that the composite map of simplicial sets N k Sd wS • C α • −−→ N k w Cob big ( C ) • τ • −→ N k wS • C , given by restriction along Ar[ n ] ⊂ Ar([ n ] op ∗ [ n ]), is the last-vertex map. This isa weak equivalence for each k ≥
0, and therefore so is also the induced compositemap of spaces (cid:12)(cid:12) N • Sd wS • C (cid:12)(cid:12) τ ◦ α −−→ (cid:12)(cid:12) N • wS • C (cid:12)(cid:12) . Similarly, we may apply the edgewise subdivision to the modified cobordism cate-gory and consider the composite simplicial functor(2) Sd w Cob big ( C ) • Sd τ • −−−→ Sd wS • ( C ) α • −−→ w Cob big ( C ) • ≃ w Cob ( C ) • . This composition is not the last-vertex map for w Cob big ( C ) • . Indeed, after iden-tifying the poset [ n ] op ∗ [ n ] with its opposite, an n -simplex in the simplicial set ofobjects of Sd w Cob big ( C ) • identifies with a diagram f Ar (cid:0) ([ n ] op ∗ [ n ]) ∗ ([ n ] op ∗ [ n ]) (cid:1) → C . Then α • ◦ Sd τ • corresponds to the restriction along the inclusion j : [ n ] op ∗ [ n ] ⊂ ([ n ] op ∗ [ n ]) ∗ ([ n ] op ∗ [ n ])into the third and fourth factors. On the other hand, the last-vertex functor(3) L : Sd w Cob big ( C ) • L −→ w Cob big ( C ) • ≃ w Cob ( C ) • corresponds to the restriction along the inclusion j : [ n ] op ∗ [ n ] ⊂ ([ n ] op ∗ [ n ]) ∗ ([ n ] op ∗ [ n ])into the first and fourth factors. However, the two inclusion functors are relatedby a (unique) natural transformation j ⇒ j , which induces a simplicial naturaltransformation(4) L ⇒ α • ◦ Sd τ • : Sd w Cob big ( C ) • → w Cob big ( C ) • ≃ w Cob ( C ) • . between the composite (2) and the last-vertex functor (3).This natural transformation (4) has an extra property: given an object G in thecategory Sd w Cob big ( C ) n , the morphism F := L ( G ) → F := ( α • ◦ Sd τ • )( G )is such that the diagram F ij / / / / (cid:15) (cid:15) F ik (cid:15) (cid:15) F ij / / / / F ik is a pushout square for each 0 ≤ i ≤ j ≤ k ≤ n . (Indeed, we have F ij = G µ ( i ) ρ ( j ) and F ij = G ν ( i ) ρ ( j ) where µ, ν : [ n ] op → [ n ] op ∗ [ n ] ∗ [ n ] op ∗ [ n ] are the two inclusions and ρ : [ n ] → [ n ] op ∗ [ n ] ∗ [ n ] op ∗ [ n ] is the inclusion into the last factor.) This meansthat the natural transformation (4) defines a simplicial functor H • : Sd w Cob big ( C ) • → w Cob po ( C ) • such that the composition with the source and target projections, gives the last-vertex functor (3) and the simplicial functor (2), respectively. By Lemma 2.3, thesource and target projections induce homotopic maps after geometric realization,therefore also the maps induced by (2) and (3) must be homotopic. Since (3)induces the last-vertex map which is a weak homotopy equivalence, so is also themap induced by (2) and the result follows. (cid:3) The classifying space
B Cob ( C , w C ) is based at the zero object of the Waldhausencategory C , denoted ∗ ∈ w C = w Cob ( C ) and regarded as a 0-simplex (‘object’) ofthe cobordism category Cob ( C , w C ). This choice of basepoint is natural with respectto exact functors of Waldhausen categories. Moreover, the natural comparison map τ preserves the basepoint. Thus, passing to the loop spaces, we obtain the followingcorollary. Corollary 3.2.
The map Ω( τ ) : Ω B Cob ( C , w C ) −→ K ( C ) is a weak homotopyequivalence. Combined with Proposition 2.10, this also yields the following corollary.
Corollary 3.3.
Let C be a Waldhausen category with functorial factorizations.Then there is a natural (zigzag) weak homotopy equivalence Ω B Cob sym ( C , w C ) ≃ K ( C ) . Deloopings.
Let C be a Waldhausen category. The application of the S • -construction can be iterated since S • C defines a simplicial object in the categoryof Waldhausen categories and exact functors. It is well known that the iteration ofthe S • -construction produces canonical deloopings of K ( C ), see [12, 1.5].The same is true for the cobordism construction C 7→
Cob ( C ) • applied to aWaldhausen category C . First, in order to see that the cobordism category con-struction can be iterated, it suffices to promote the simplicial category Cob ( C ) • toa simplicial object in the category of Waldhausen categories. As subcategory ofcofibrations co Cob ( C ) n ⊂ Cob ( C ) n , we consider the subcategory of those naturaltransformations φ : ( F •• ) → ( F ′•• ) such that for each 0 ≤ i ≤ n , the morphism offiltered objects in F n − i +1 C F ii / / / / φ ii (cid:15) (cid:15) F i,i +1 / / / / φ i,i +1 (cid:15) (cid:15) · · · / / / / F inφ in (cid:15) (cid:15) F ′ ii / / / / F ′ i,i +1 / / / / · · · / / / / F ′ in is a cofibration in F n − i +1 C (see [12, 1.1]). This subcategory of cofibrations togetherwith w Cob ( C ) n makes Cob ( C ) n into a Waldhausen category which is natural in[ n ] ∈ ∆ op . Thus, the cobordism category construction can be iterated so that weobtain an n -fold simplicial category, for n ≥ C 7→
Cob ( n ) ( C ) •···• . We write B ( n ) Cob ( C , w ) for the classifying space of this multisimplicial category(i.e., the geometric realization of the associated n -fold simplicial space). COBORDISM MODEL FOR WALDHAUSEN K -THEORY 15 Let
Cob ( C ) , ∗ ⊂ Cob ( C ) denote the full subcategory which consists of thosediagrams ( F ij ) such that F = F = ∗ . Then we have Cob ( C ) , ∗ = C . Each pointin w Cob ( C ) , ∗ defines a loop in B Cob ( C , w ) and therefore we have a natural map(5) (cid:12)(cid:12) w C (cid:12)(cid:12) → Ω B Cob ( C , w C ) . Using the identification τ , this agrees with the usual “group completion” map (cid:12)(cid:12) w C (cid:12)(cid:12) → K ( C ). Moreover, using the naturality of this map (5), we also obtainnatural maps(6) B ( n − Cob ( C , w C ) → Ω B ( n ) Cob ( C , w )which make the sequence of spaces { B ( n ) Cob ( C , w ) } n ≥ into a spectrum. As aconsequence of Theorem 3.1 and the naturality of τ , we obtain natural (zigzag)weak homotopy equivalences(7) Ω n B ( n ) Cob ( C , w ) ≃ Ω n (cid:12)(cid:12) N • wS ( n ) • C (cid:12)(cid:12) . These maps are also natural in n ≥
1, so they define a (zigzag) map of spectra.Using the fact that iterating the S • -construction defines canonical deloopings [12,1.5], it follows that the maps (6) are also weak homotopy equivalences. As a con-sequence, the spectrum { B ( n ) Cob ( C , w C ) } n ≥ is an Ω-spectrum.4. Example: Cobordism categories and A -theory Models for A -theory. Let X be a topological space and R ( h ) f ( X ) the Wald-hausen category of relative (homotopy) finite retractive space over X (see [12, 2.1]).It is well known that R ( h ) f ( X ) has functorial factorizations given by a mappingcylinder construction.We now consider the (symmetric) cobordism categories associated to these twoWaldhausen categories. Explicitly, in the case of R hf ( X ), the symmetric cobordismcategory Cob sym ( R hf ( X ) , w ) is a semi-simplicial space such that:(i) the space of 0-simplices is the moduli space of relative homotopy finite re-tractive spaces over X (with respect to the class of homotopy equivalences),(ii) the space of 1-simplices is the moduli space of diagrams in R hf ( X ) asfollows: Y Y > > > > ⑤⑤⑤⑤⑤⑤⑤⑤ Y a a a a ❇❇❇❇❇❇❇❇ such that Y ∪ X Y → Y is a cofibration (with respect to the objectwisehomotopy equivalences between such diagrams of spaces). Such a diagrammay be regarded as a formal cobordism between the retractive spaces Y and Y .(iii) the space of n -simplices is the space of n -composable strings of 1-simplices.Theorems 3.1 and 3.3 imply that there are natural (zigzag) homotopy equivalencesin X Ω B Cob sym ( R hf ( X ) , w ) ≃ Ω B Cob ( R hf ( X ) , w ) ≃ A ( X ) = K ( R hf ( X )) . There are similar homotopy equivalences in the case of R f ( X ). More generally, for a fibration p : E → B , we may consider the cobordism cat-egory Cob sym ( R hf ( p ) , w ) associated with the Waldhausen category of R hf ( p ) (see[7]). Then we obtain similarly natural (zigzag) homotopy equivalences in p ,(8) Ω B Cob sym ( R hf ( p ) , w ) ≃ Ω B Cob ( R hf ( p ) , w ) ≃ A ( p ) = K ( R hf ( p )) , where A ( p ) denotes the bivariant A -theory of p .Interestingly, the additional flexibility of working with unpointed Waldhausencategories furnishes yet another model for A -theory. Given a space X , let C ( h ) f ( X )denote the category whose objects are pairs ( Y, u : Y → X ) where Y is (homotopyequivalent to) a finite CW complex. A morphism f : ( Y, u ) → ( Y ′ , u ′ ) in C hf ( X ) isa map f : Y → Y ′ such that u = u ′ f . We say that this is a cofibration (resp. weakequivalence) if the underlying map f is a Hurewicz cofibration (resp. homotopyequivalence). In the case of C f ( X ), the cofibrations are the inclusions of CWcomplexes. Note that the category C ( h ) f ( X ) has an initial object given by the pair( ∅ , ∅ → X ). With this structure, C ( h ) f ( X ) becomes an unpointed Waldhausencategory with initial object and functorial factorizations.The correspondence X
7→ C ( h ) f ( X ) defines a functor from spaces to the categoryof unpointed Waldhausen categories which sends a map f : X → X ′ to the exactfunctor C ( h ) f ( X ) → C ( h ) f ( X ′ ) , ( Z, Z → X ) ( Z, Z → X → X ′ ) . The associated symmetric cobordism category
Cob sym ( C hf ( X )) • may be regarded asthe category of formal cobordisms between homotopy finite spaces with a structuremap to X . By Proposition 2.10, we also have natural homotopy equivalences in X (9) B Cob sym ( C ( h ) f ( X ) , w ) ≃ B Cob ( C ( h ) f ( X ) , w ) . There are exact functors of unpointed Waldhausen categories relating C hf ( X )and R hf ( X ). First, there is an exact functor that adds a disjoint copy of X ,J X : C hf ( X ) → R hf ( X )( Y, u : Y → X ) ( Y ⊔ X, X ⊆ Y ⊔ X u +id X −−−−→ X ) . Secondly, when X is homotopy finite, there is an exact functor that forgets thesection, U X : R hf ( X ) → C hf ( X )( Z, X Z r −→ X ) ( Z, r : Z → X ) . Note that the last functor does not preserve the initial object – but it preservespushouts. These functors restrict also to exact functors on C f ( X ) and R f ( X )respectively when X is a finite CW complex. Proposition 4.1.
Let X be a homotopy finite space. Then the exact functors U X and J X induce inverse homotopy equivalences: Ω B Cob (J X , w J X ) : Ω B Cob ( C hf ( X ) , w C hf ( X )) → Ω B Cob ( R hf ( X ) , w R hf ( X ))Ω B Cob (U X , w U X ) : Ω B Cob ( R hf ( X ) , w R hf ( X )) → Ω B Cob ( C hf ( X ) , w C hf ( X )) . Moreover, the same is true for the restrictions of these functors to C f ( X ) and R f ( X ) respectively, when X is a finite CW complex. COBORDISM MODEL FOR WALDHAUSEN K -THEORY 17 Proof.
By Proposition 2.6, it suffices to show that both composite functors U X ◦ J X and J X ◦ U X are connected to the respective identity functors by relative isomor-phisms. For the composite U X ◦ J X : C hf ( X ) → C hf ( X ),( Y, u : Y → X ) ( Y ⊔ X, Y ⊔ X u +id X −−−−→ X ) , there is a relative isomorphism φ : Id → U X ◦ J X where φ ( Y,u ) : Y → Y ⊔ X is the canonical inclusion. For the composite J X ◦ U X : R hf ( X ) → R hf ( X ), givenby ( Y, X Y r −→ X ) ( Y ⊔ X, X ⊆ Y ⊔ X r +id X −−−−→ X ) , there is a relative isomorphism ψ : J X ◦ U X → Id where ψ ( Y,X i Y r −→ X ) : Y ⊔ X id Y + i −−−−→ Y. The same argument applies for the comparison of C f ( X ) and R f ( X ). (cid:3) Remark . Similarly we can define an unpointed Waldhausen category C hf ( p ) fora fibration p : E → B . The classifying space of its cobordism category is a modelfor A ( p ) when p has homotopy finite fibers. The proof is exactly the same. Proposition 4.3.
The exact inclusion functor C f ( X ) → C hf ( X ) (of unpointedWaldhausen categories) induces a homotopy equivalence B Cob ( C f ( X ) , w C f ( X )) ≃ −→ B Cob ( C hf ( X ) , w C hf ( X )) . Proof.
It is well known that the inclusion functor R f ( X ) → R hf ( X ) induces a K -equivalence as a consequence of Waldhausen’s Approximation Theorem (see [12,Proposition 2.1.1]). (Therefore the statement of the proposition for a finite CWcomplex X follows directly from Proposition 4.1.) The argument for the inclusionfunctor C f ( X ) → C hf ( X ) is similar so we only sketch the proof. For each n ≥ Cob sym ( C f ( X )) n → Cob sym ( C hf ( X )) n satisfies the conditions of the Approximation Theorem in [12, pp. 35-37]. For n = 0,the argument is similar to [12, Proposition 2.1.1], and for n >
0, the proof of [12,Lemma 1.6.6] is easily adapted to this purpose. Then, following the proof of theApproximation Theorem in [12], or applying [3, Lemma 7.6.7], we conclude thatthe functor w Cob sym ( C f ( X )) n → w Cob sym ( C hf ( X )) n induces a homotopy equivalence after passing to the classifying spaces – as shownin the proof of [3, Lemma 7.6.7], only the existence of an initial object, rather thana zero object, is required. The result then follows. (cid:3) While U X is well defined only when X is homotopy finite and it is not naturalin X , the functor J X is a natural transformation defined for any X . Then we havethe following result which generalizes Proposition 4.1 to an arbitrary X . Theorem 4.4.
The natural map Ω B Cob ( C hf ( X ) , w C hf ( X )) Ω B Cob (J X ,w J X ) −−−−−−−−−−−→ Ω B Cob ( R hf ( X ) , w R hf ( X )) ≃ A ( X ) is a weak homotopy equivalence for any space X . Proof.
The functor X
7→ | w Cob ( C hf ( X )) n | preserves homotopy equivalences be-cause it sends the endpoint inclusions i , i : X → X × I to homotopic maps - eachof these is homotopic to the map induced by the functor( Z, u : Z → X ) ( Z × I, u × id : Z × I → X × I ) . As a consequence, the functor(10) X B Cob ( C hf ( X ) , w C hf ( X ))preserves homotopy equivalences, too. We claim that the functor (10) also preservesweak homotopy equivalences. Fix a functorial CW-approximation g X : X c ∼ −→ X .Then there is a functorΦ n : w Cob ( C hf ( X )) n → w Cob ( C hf ( X c )) n which is given by applying the functorial CW-approximation and replacing the mapsin the cospans functorially by cofibrations as necessary. The composite functor w Cob ( C hf ( X )) n Φ n −−→ w Cob ( C hf ( X c )) n g X ∗ −−→ w Cob ( C hf ( X )) n is weakly equivalent to the identity functor. The other composite functor(11) w Cob ( C hf ( X c )) n g X ∗ −−→ w Cob ( C hf ( X )) n Φ n −−→ w Cob ( C hf ( X c )) n is described as follows: (i) first, it applies the CW-approximation functor (again),then (ii) it replaces the maps in the cospans by cofibrations as necessary, and lastly,(iii) it composes the induced structure map to ( X c ) c with the map g cX : ( X c ) c → X c .This last map is homotopic to g X c : ( X c ) c → X c and therefore after applying | w Cob ( C hf ( − )) n | , these two maps g cX and g X c induce the same map up to homo-topy. Thus, using in (iii) the map g X c instead, we obtain a composite functor whichis weakly equivalent to the identity functor and it induces a map homotopic to theone induced by (11). It follows that (10) sends g X to a homotopy equivalence. Asa consequence, the functor (10) preserves weak homotopy equivalences. It is wellknown that the A -theory functor has this property too (see [12, Proposition 2.1.7]).Then it suffices to prove that Ω B Cob (J X , w J X ) is a weak equivalence when X is a CW complex. We may write X as the (homotopy) filtered colimit of its finitesubcomplexes. We note that the composite functor Cob ( C f ( − ) , w C f ( − )), definedon objects by X
7→ C f ( X ) w Cob ( C f ( X )) • B Cob ( C f ( X ) , w C f ( X )) , preserves this filtered colimit and therefore, after applying Proposition 4.3, it fol-lows that the functor (10) preserves this (homotopy) filtered colimit up to weakequivalence. Using the Waldhausen category R f ( X ) for the definition of A -theory,the analogous argument shows that the same is true for the A -theory functor. Thenthe result follows by naturality and Proposition 4.1. (cid:3) The map from the cobordism category of manifolds.
Using the cobor-dism model for A -theory, we give a new description of the map from the cobordismcategory of [4, 5] to A -theory, which was defined in [2] and studied further in [7, 8].Here we will focus on the description of the map presented in [8].Let C ∂ ( θ ) denote the cobordism (non-unital) category of θ -structured manifoldswith boundary as defined in [8, Section 2], where θ : X → BO ( d ) is the fibration COBORDISM MODEL FOR WALDHAUSEN K -THEORY 19 which determines the tangential structure. For fixed θ : X → BO ( d ), there is asemi-simplicial map N • C ∂ ( θ ) → Ob Cob sym ( C hf ( X )) • J X −−→ Ob Cob sym ( R hf ( X )) • which sends an embedded θ -structured cobordism ( W ; M , M ) to the cospan W ⊔ XM ⊔ X rrrrrrrrrr M ⊔ X e e e e ▲▲▲▲▲▲▲▲▲▲ of retractive spaces over X (cf. [7, Section 5.1], [8, Section 5.1]). Thus, this map cor-responds essentially to an inclusion of cobordisms of compact smooth manifolds into(formal) cobordisms of homotopy finite spaces. We note that this semi-simplicialmap preserves the basepoint that is defined by the initial object. After geometricrealization, we obtain a map of spaces τ ( θ ) : Ω B C ∂ ( θ ) → Ω B Cob sym ( C hf ( X ) , w C hf ( X )) ∼ −→ Ω B Cob sym ( R hf ( X ) , w R hf ( X )) . In this map, the topology on the cobordism category is not yet encoded. Thiscan be rectified, just as in [8], by introducing an additional simplicial direction toobtain the simplicial thickening C ∂ ( θ ) • , a simplicial object with values in (non-unital) categories (see [8, Section 2]). Then the definition of the map above appliessimilarly in each simplicial degree and produces, as in [8, Section 5.1], a simplicialmorphism τ ( θ ) • to the simplicial thickening (or thick model ) of the symmetriccobordism category associated to R hf ( X ),[ n ] Ω B Cob sym R hf X × ∆ n ↓ ∆ n , w R hf X × ∆ n ↓ ∆ n . The passage to the (geometric realization of the) simplicial thickening does notchange the homotopy type of this cobordism category of retractive spaces. Thiscan be seen either by following the arguments of [8], or by using the equivalencewith bivariant A -theory and the analogous statement in this setup from [2], [7,Section 3.3].Using the identification of Theorem 3.1, the (simplicial thickening of the) map τ ( θ ) is precisely the map to A -theory as defined in [8, Section 5.1].Note that the summand X in the cospan above can be omitted by working insteadwith the simplicial unpointed Waldhausen category C hf ( X ) in order to model A ( X )(Theorem 4.4). References [1] C. Barwick,
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G. RaptisFakult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany
E-mail address : [email protected] W. SteimleInstitut f¨ur Mathematik, Universit¨at Augburg, D-86135 Augsburg, Germany
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