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N simp ,a f, , | elet we , eeaie inrsclassifying Milnor’s generalizes respectively with ) . or Top enn,tecomposi- the meaning, ; | , Ner a OR C n opsto of composition and 1 and 0 = fat soitdt it: to associated · C stesubcategory the is eteone-object the be . fat · | ,i f, so the of ts ia groupoid gical C scanonically is f Sg4.Its [Seg74]. nalgebraic in rde the bridges d This id. 6= ≤ fsimpli- of k ory -spaces, i with ) ment G, Segal’s construction always gives us the classifying space of numerable G- structures. In fact, by the construction, the space | Ner GN| admits a numerable universal G-structure [Bot72, Appendix]. On the contrary, the geometric real- ization of the nerve of a topological groupoid does not always give us the right homotopy type of the classifying space, for example, a topological group that is homeomorphic to the Cantor set; we learn this example from A. Henriques on MathOverflow. However, if replacing geometric realization with fat realiza- tion, we get the right homotopy type of the classifying space [HG]. The third construction Csimp is of importance in model category theory as it computes homotopy colimit [Hir02], with respect to the projective model structure on the category of simplicial spaces. Also, it is a useful tool in proving theorems (e.g. [Wal83]). Comparison theorems between these constructions allow us to choose models appropriate to different problems and help us understand the geomet- ric meanings of homotopy-theoretic constructions—for instance, the geometric meaning of the algebraic K-theory space of a structured category, e.g. an exact category or a Waldhausen category internal in Top. These constructions can be generalized to simplicial spaces or even simplicial objects in a topologically (simplicially) enriched model category M. Further- more, given a simplicial object X· in M, they are connected by the natural morphisms N π fat q l simp X· −→ X· −→ X· ←− X· . (1) The map l is the Bousfield-Kan map (or the last vertex map); in the case of simplicial spaces, it has been studied in [BK72], [Seg74], and in the case of simplicial objects in M, [Hir02], [Dug]. The map q, the canonical quotient map from the fat realization to the geometric realization, is rather well understood; [Seg74] and [tD74] treat the case of simplicial spaces and [Dug] the case of simplicial objects in M. On the contrary, the map π is less studied, and [tD74] is the only reference dealing with the map π that we can find in the literature. [tD74, Proposition 2] asserts that the map π is a homotopy equivalence in the case of simplicial spaces, but it appears that the proof contains some gaps (see Remark 5.2); nevertheless, it remains a very promising assertion. In this paper, we present a unified approach, due to Segal, that allows us to compare these constructions simultaneously; the approach is based on an associative law implicitly used in [Seg74, Appendix A] and the Reedy model structure on sM, the category of simplicial objects in M. With this approach, we obtain two comparison theorems that recover and generalize most of the known results we know of concerning the relation between these four realization functors, and in particular, we obtain a complete proof of a generalized tom Dieck theorem. Our approach relies heavily on a generalized Segal lemma ([Seg74, Lemma A.5]) for Top-enriched model categories. Theorem 1.1. Let sM and cTop be the Reedy model categories of simplicial objects in M and cosimplicial spaces, respectively, and suppose the morphisms · · · f· : X· → Y· ∈ sM and g : I → J ∈ cTop are level-wise weak equivalences between cofibrant objects. Then the induced map between the associated coends

△ △ △ f g· : X I· → Y J · Z · Z · Z · is a weak equivalence in M, where  is the tensor product functor from M×Top

2 △ to M, and (−)(−) ∈M denotes the coend of a simplicial object in M and a cosimplicialR space. △ is the simplex category. The category Top admits at least three different model structures [MP12] and the theorem applies to all of them. However, for our purpose, we are primarily concerned with the Strøm model structure. The simplicial version of Theorem 1.1 is discussed in details in [Hir02, 18.4]; our proof is different from [Hir02] and based on the decomposition of latching objects in [GJ99, VII]. Top- (s Sets-)enriched model categories of interest to us are the category Top [MP12], the category of simplicial sets s Sets [GJ99], the category of (simplicial) spectra or Γ-spaces [BF78], and the category of chain complexes [MP12].

1.1 Main Theorems

Theorem 1.2. For any cofibrant object X· in sM, the natural morphisms

△ △ △ △ π q l XN△· −→ Xfat△· −→ X △· ←− Xsimp△· Z · Z · Z · Z · are weak equivalences in M, where the cosimplicial space △· is given by the n geometric realization of the standard n-simplex △· in s Sets. The theorem implies that the geometric realizations of the simplicial spaces in (1) are homotopy equivalent when X· is a proper simplicial space.

Theorem 1.3. Given a level-wise cofibrant object X· in sM, the following coends △ △ △ π XN△· −→ Xfat△· ≃ Xsimp△· Z · ≃ Z · Z · are weakly homotopy equivalent—connected by a zig-zag of weak equivalences. Applying the theorem to the Strøm model category Top, we see the projection

N fat π : ||X· × S·|| = |X· | → |X· | = ||X·|| (2)

is a homotopy equivalence, for every simplicial space X·, and hence recover [tD74, Proposition 2], where S· is the semi-simplicial set defined by Sn := {i0 < ... < in | ij ∈ N}. The idea of the proof comes from [Seg74, p309-310], where a kind of associativity is implicitly employed—we interpret it as an associative law in infinite-dimensional linear algebra, namely

(vT A)w = vT (Aw),

for any ∞-by-∞ matrix A and column vectors v and w. △ fat · △ N · In the last section, we define a map τ : X· △ → X· △ to replace the map ρ constructed in [tD74, p.47]1 andR prove that, underR the same condition of Theorem 1.3, the map τ is a homotopy inverse to the map π. The author wishes to thank Sebastian Goette for suggesting the construction of the map τ. He gratefully acknowledges use of facilities at and the financial support from Mathematical Research Institute of Oberwolfach. He thanks God, who gives him life and sustains him.

1The construction of ρ appears not to give a well-defined map (see Remark 5.2).

3 2 Left Kan extension

Let △+ denote the subcategory of the simplex category △ consisting of injective op morphisms. Then a semi-simplicial object in M is a functor X· : △+ → M and its left Kan extension LX·, with respect to the inclusion △+ ֒→△, is given by LXn := Xk, av [n]։[k] where ։ (resp. ֌) stands for a surjective (resp. injective) morphism [ML71, Chapter X], [tD74, p.42]. The simplicial structure of LX· can be described as u su,v iu,v follows: Given a morphism [n′] −→ [n], we let [n′] ։ [k′] ֌ [k] be the unique v ′ u ∗ factorization of the composition [n ] −→ [n] ։ [k]. Then u : LXn → LXn′ is given by ∗ ` iu,v v Xk −−−−→ Xk′ . av sau,v [n]։[k] [n′] ։ [k′] The following generalizes [tD74, Lemma 1].

Lemma 2.1. Given a semi-simplicial object X·, there is a canonical isomor- phism △ △ + ∼ X △· −→= LX △·. Z · Z · Proof. The isomorphism is given by the inclusion

n ` id n n Xn△ −−−→ Xn△ ⊂ Xm△ . a[n] a[n] a[n] = ։ [n]−→[n] [n] [m]

To define its inverse, we observe that the following morphisms

` id v∗ n [n];v s Xs△ −−−−−→ Xs△ ; a[n] a[s] v [n]։[s]

` id su,v,∗ n u,v s′ Xs△ −−−−−−−−→ Xs△

au aiu,v [n]−→[m] [s′] ֌ [s] v [m]։[s]

respect face and degeneracy maps in LX· and face maps in X·, and hence they induce a morphism △ △+ LX △· → X △·. Z · Z ·

4 Example 1: Let X· be a simplicial object in M. Regarding it as a semi- simplicial object by the inclusion △+ ֒→△, we denote its left Kan extension by fat fat X· , and there is a canonical projection X· → X· given by the assignment

` u∗ u Xm −−−→ Xn. au an [n]։[m]

Example 2: Given an object X· in sM, we denote the left Kan extension of N N the semi-simplicial object X· × S· (2) by X· , whose n-th component Xn can be described as follows: Xk. [n]։a[k]֌N

N fat There is a canonical projection from X· → X· given by

` id Xk −−−→ Xk [n]։a[k]֌N [na]։[k]

The constructions (−)N and (−)fat can be viewed as functors from sM to it- self, and they generalize Segal’s unraveling construction and the fat construction defined in the introduction; namely, the following diagrams are commutative

N (−) (−)fat CatM CatM CatM CatM (3) Ner· Ner· Ner· Ner·

N (−) (−)fat sM sM sM sM

where CatM is the category of internal categories in M. Lemma 2.2. The canonical maps of simplicial sets

n,N π n,fat q n l simp △· −→△· −→△· ←−△·

induce homotopy equivalences

N π q l △n, −→△n,fat −→△n ←−△simp

after geometric realization.

n Proof. Since the standard n-simplex △· is the nerve of the linearly ordered set [n] := {0 ≤ 1... ≤ n}, it suffices to show that the following functors

N π q l [n] −→ [n]fat −→ [n] ←− [n]simp

induce homotopy equivalences, where [n]simp is the over category △ ↓ [n]. It is clear that [n]simp have the terminal object n = n, and the object n is the terminal object in [n]fat and [n]. Hence, if we can show that the composition

5 q ◦ π :[n]N → [n] induces a homotopy equivalence, then the lemma follows. To see this, we define an intermediate category [n]N,′ of [n]N which consists of objects (k,l) with k ≤ l and observe that q ◦ π can be decomposed into two natural projections

N N,′ π1 :[n] → [n] , (k,l) 7→ (k, k) k ≥ l, (k,l) 7→ (k,l) k ≤ l; N,′ π2 :[n] → [n] (k,l) 7→ k.

It is clear that the slice category π1 ↓ (k,l) has a terminal object (k,l) and the slice category π2 ↓ k has an initial object (0, 0). By Quillen’s theorem A, the functors π1, π2 induce homotopy equivalences, and hence the proof is complete.

3 An associative law

Lemma 3.1. There are canonical isomorphisms

△ △ ∼ X △·,N −→= XN△· Z · Z · △ △ ∼ X △·,fat −→= Xfat△· Z · Z · △ △ ∼ X △·,simp −→= Xsimp△· Z · Z ·

In the case of simplicial spaces, the last two isomorphisms has been implicitly used in [Seg74, p.309] and [Wal83, p.359], and a detailed explanation of the second isomorphism using the universal property of Kan extension is given in [tD74, Lemma 1]. Here, we present a unified approach to such isomorphisms, viewing them as a consequence of an associative law in infinite dimensional linear algebra. Proof. Firstly, we observe that the assignments

∗  n,N ` u N Xn △k = Xn −−−→ Xr = Xk , au ։a N →֒[r]−→[n] [k] [r] k]։[r]֒→N] ∗  n,fat ` u fat Xn △k = Xn −−−→ Xr = Xk , au a։ [r]−→[n] [k] [r] [k]։[r] ∗  n,simp ` u simp Xn △k = Xn −−−→ Xrk =: Xk

[r0a]→... [r0]→a...→[rk] u →[rk]−→[n]

6 induce the isomorphisms, whose inverses are given by the obvious inclusions,

△ N ∼ N X △·, −→= X Z · · · △ ∼ X △·,fat −→= Xfat (4) Z · · · △ ∼ X △·,simp −→= Xsimp. Z · · ·

Then, as different ways of computing colimits yield the same result, there is an isomorphism △ △ △ △ ·,− · ·,− · X·( △ △ ) =∼ ( X·△ )△ , (5) Z Z · Z Z · where − can be N, fat or simp. By (4), isomorphism (5) gives the isomorphism

△ △ ·,− − · X·△ =∼ X △ . Z Z ·

n,N n,fat n,simp Remark 3.2. If we view (△k ), (△k ), and (△k ) as matrices and (Xn) and (△k) column vectors, then (5) resembles an associative law in linear algebra.

4 Comparison theorems

This section discuss a generalized version of Segal’s lemma [Seg74, Lemma A.5] for a Top-enriched model category M; an analogous version for simplicially enriched model categories can be found in [Hir02, Corollary 19.4.13-14].

· · · Lemma 4.1. Let f· : X· → Y· and g : I → J be level-weak equivalences between cofibrant objects in sM and cTop, respectively. Then the induced map between the associated coends

△ △ △ f g· : X I· → Y J · Z · Z · Z · is a weak equivalence in M. Proof. The idea has been sketched in [Seg74, Appendix] (see also [tD74, p.43], [Dug, p.11], [GJ99, p.375]). For the sake of completeness, we give a detailed proof here. Firstly, we claim that the latching object of any cofibrant object in sM (resp. cTop) is cofibrant. Following [GJ99, p.362-6], we consider the category On whose objects are surjective morphisms [n] ։ [m] with n>m and morphisms from [n] ։ [m] to [n] ։ [m′] are those morphisms [m] → [m′] satisfying the commutative diagram [n]

[m] [m′]

7 The n-th latching object of a (co)simplicial object X·, denoted by LnX·, is then given by the colimit X . colim opm [n]։[m]∈On

Now, for each 1 ≤ k ≤ n, we can define the subcategories of On

Mn,k := {φ :[n] ։ [m] | φ(k) ≤ k} M(k − 1) := {φ :[n] ։ [m] | φ(k − 1) = φ(k)}.

Let Ln,kX· be the colimit colimt Xm. ։ [n] [m]∈Mn,k

Then there is a filtration of LnX· given by

Xn−1 = Ln,1X· ⊂ Ln,2X· ⊂ ... ⊂ Ln,nX· = LnX· (6)

and a pushout diagram

∗ sk Ln−1,kX· colim Xm = Xn−1 [n]։[m]∈M(k)

∗ (7) sk

Ln,kX· Ln,k+1X·

where sk :[n] ։ [n − 1] is the degeneracy map with sk(k) = sk(k +1). By induction and pushout diagram (7), we see filtration (7) is a sequence of cofi- brations and the object Ln,kX· is cofibrant, for every n, k. In particular, the n · n · objects L I , L J , LnX·, and LnY· are cofibrant, for every n. Now, since M is a Top-enriched category, given a cofibrant object Z in M and a cofibrant object A in Top, the functors below

A− : M 7→ M −Z : Top 7→ M preserve cofibrations and weak equivalences between cofibrant objects [MP12, Lemmas 14.2.9; 16.4.5]. Thus, we have the following weak equivalences between two cospan of cofibrations

n · n · n n · n · n (XnL I ← LnX·L I → LnX·I ) → (YnL J ← LnY·L J → LnY·J ). (8) Since the subcategory of cofibrant objects in a model category is always proper, (8) induces a weak equivalence between the pushouts

n · n n · n  n ·   n ·  Xn L I ∪LnX·L I LnX· I → Yn L J ∪LnY·L J LnY· J . (9)

Now, recall that the n-skeleton object (skn Z·)· of a simplicial object is defined by first truncating Z· at the n-th degree, denoted by Z¯· and then freely throwing the degeneracies [GJ99, p.354], namely

[m] 7→ (skn Z·)m := colim Z¯k. [m]։[k] k≤n

8 △ · By the definition, we have colim(skn Z·)· = Z·. Since the n-skeleton (skn X·)·I n △ · R (resp. (skn Y·)·J ) is the pushout of the cospan R △ · n · n n (sk X ) I ← X L I ∪  n · L X I → X I Z n−1 · · n LnX· L I n · n

△ · n · n n (resp. (sk Y ) J ← Y L J ∪  n · L Y J → Y J ), Z n−1 · · n LnY· L J n · n and the second arrow in each span is a cofibration—M is Top-enriched [MP12, Lemma 16.4.5], by induction, we get the weak equivalence

△ △ sk X I· → sk Y J ·, Z n · Z n ·

△ for every n. The theorem then follows from the fact that the functor (−)I· △ (resp. (−)J ·) is a left adjoint and commutes with colimits. R R As a corollary of Lemmas 2.2, 3.1, and 4.1, we have the following theorem (compare with [Hir02, Theorem 18.7.4], [Seg74, Proposition 1], [tD74, Proposi- tion 1]).

Theorem 4.2. Given a cofibrant object X· in sM, the canonical maps

△ △ △ △ π q l XN△· −→ Xfat△· −→ X △· ←− Xsimp△· Z · Z · Z · Z · are weak homotopy equivalences. The following can be deduced from Theorem 4.2.

Theorem 4.3. Let X· be a level-wise cofibrant object in sM. Then the objects below are weakly equivalent

△ △ △ N · π fat · simp · X· △ −→ X· △ ≃ X· △ . Z ≃ Z Z

Proof. Let Y· be a cofibrant replacement of X·. Then the theorem ensues from the following commutative diagram of weak equivalences

△ N · △ fat · △  · △ simp · Y· △ Y· △ Y· △ Y· △ R R R R

△ N · △ fat · △ simp · X· △ X· △ X· △ R R R The horizontal arrows above are weak equivalences by Theorem 4.2. The vertical N N fat arrows are weak equivalences by Lemma 4.1 because Y· → X· (resp. Y· → fat simp simp X· and Y· → X· ) is a level-wise weak equivalence between cofibrant objects in sM—the assumption that X· is level-wise cofibrant is used here.

9 △ simp · Remark 4.4. The coend X· △ computes the homotopy colimit [Dug, Part I.4], [Hir02, Chapter 18R .1] of the diagram X·, and hence, all these variants of geometric realization in Theorem 4.3 compute the homotopy colimit of a level- wise cofibrant object in sM (compare with [Dug, Section 17.4]). However, when viewing the homotopy colimit functor as the left Kan extension of the colimit functor, we do not use the Reedy model structure on sM but the projective one [Hir02, Section 11.6], [Dug, Section 5.8].

5 The homotopy inverse to π

n Let Sd· △ be the semi-simplicial set given by n Sdk △ := {[l0] ֌ ... ֌ [lk] ֌ [n] | li

n Then the fat realization of Sd· △

△+ Sd △n ×△· Z · is the barycentric subdivision of △n and canonically homeomorphic to △n. Now, consider the assignment below

n n m k n k τ¯ : △m × Sdk △ ×△ → △k × Sk ×△ m,ka ak ∗ (x, [l0] ֌ [l1] ֌ ... ֌ [lk] ֌ [m],t) 7→ (u x, l0

[li] ֌ ... ֌ [lk] ֌ [m], and observe that the assignment descends to a map of coends

△+ △+ △+ τ n : △n,fat = △n × Sd △· ×△· → (△n × S ) ×△· = △n,N, Z · Z · Z · · where the second identity follows from Lemma 2.1. Observe also that the com- position τ n π △n,fat −−→△n,N −→△n,fat is induced by the standard map from the barycentric subdivision of a simplex to itself, and hence the linear homotopy gives a homotopy of cosimplicial spaces H : △·,fat × I →△·,fat

n between the identity and π ◦τ . Now, let X· be a simplicial object in M. Then, by Lemma 3.1, the cosimplicial map τ · induces a morphism

△ △ △ △ N N τ : Xfat△· = X △·,fat → X △·, = X △·, Z · Z · Z · Z · and the homotopy H shows that the morphism τ is the homotopy inverse to π when X· is level-wise cofibrant (Theorem 4.3). Furthermore, the morphism τ is functorial with respect to X· as we have the commutative diagram below, for any morphism X· → Y· in sM,

10 τ △ ·,fat △ ·,N X·△ X·△ R R

τ △ ·,fat △ ·,N Y·△ Y·△ R R Hence, we have proved the following theorem.

Theorem 5.1. Given a simplicial object X· in M, there is a well-defined mor- phism △ △ τ : Xfat△· → XN△· Z · Z · such that the composition π ◦ τ is left homotopy to id and τ is functorial with respect to X·. If, in addition, X· is level-wise cofibrant, then the morphism τ is a homotopy inverse to the morphism π. Remark 5.2. To define a homotopy inverse to the map π in the case of sim- plicial spaces, [tD74, p.45-7] considers the following assignment

n n Xn ×△ → Xn × Sn ×△ (10)

(y; t0, ..., tn) 7→ (y, 1 < ... < n; s1,n(t0, ..., tn), ..., sn,n(t0, ..., tn)),

where sj,n(t0, ..., tn) := (j + 1) max(0, min tj − max tj ) j∈E j6∈E XE

and E runs through all subsets of [n] with j +1 elements—the map sj,n is a kind of folding map sending all the n-simplices in the barycentric subdivision of an n-simplex to the n-simplex. However, the assignment does not respect face maps—the first and second components in assignment (10) should depend on n (t0, ..., tn), the coordinate of points in △ . The map ρ is used in cohomology theories of spaces with two topologies [Mos76, Theorem 7.1].

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