Manos Lydakis 1. INTRODUCTION

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1 HOMOTOPY LIMITS OF CATEGORIES Manos Lydakis September 28, 1993 ABSTRACT We show that the homotopy limit of a diagram of nerves of categories is again a nerve of a category. We use this to prove a cofinality theorem for homotopy limits of nerves of categories. 1. INTRODUCTION Let S denote the category of simplicial sets, CAT the category of small categories, and N: CAT S the nerve functor. In [6] Thomason constructs, for any diagram of (small) categories A: I CAT, a category I A whose nerve has the homotopy type of the homotopy colimit of NA (here NA is the diagram of simplicial sets given by the composition I A CAT N S). We show that the corresponding result for homotopy limits is simpler, in the sense that the homotopy limit of NA is (naturally isomorphic to the nerve of) a category. As an application, we prove a weaker form of the Cofinality Theorem of Bousfield-Kan (Thm. XI.9.2. of [1]), but without the hypothesis that the simplicial sets involved are fibrant. This paper is an extract from the author s thesis (see 6 of [3]). We establish some notational conventions: If C and C are objects of a category C, we denote the set of maps in C from C to C by C(C, C ). If A and B are categories (and A is small) then B A denotes the category of functors from A to B (and natural maps between them). A functor F : A B gives a functor F : A C B C taking the map φ: g h to the map F φ: F g F h, and a functor F : C B C A taking the map φ: g h to the map φ F : gf hf. We write [n] for the ordered set { n} viewed as an object of CAT, and we write for the full subcategory of CAT with objects all [n]. Thus simplicial sets are functors from op to sets, and CAT ([n], C) is the set of n-simplices of the nerve of a small category C. For any map f: [m] [n] in, the induced map N n C N m C is f : CAT ([n], C) CAT ([m], C). We remark that if we write N for the nerve functor used in [1] and [6], then N C = NC op. 2. HOMOTOPY LIMITS OF CATEGORIES Let I be a small category. Recall from [1, XI.3.1] the functor Hom: (S I ) op S I S, where Hom n (X, Y ) = S I (X n, Y ). Let A and B be functors from I to CAT. Then: Hom n (NA, NB) = S I (NA n, NB) = S I (NA N[n], NB) S I (N(A [n]), NB) CAT I (A [n], B) Recall from [5] that a simplicial set is (isomorphic to) a nerve, if and only if it takes cartesian squares in op to cartesian squares in SET, and notice that the simplicial set [n] CAT I (A [n], B) has that property. Say CAT I (A [n], B) N n C. We denote this C by Hom(A, B). More precisely, define the category Hom(A, B) by ob Hom(A, B) = CAT I (A, B) mor Hom(A, B) = CAT I (A [1], B) 1

2 with the obvious structure maps. Thus Hom(N A, N B) is naturally isomorphic to NHom(A, B). If A is a functor from a small category I to CAT, we define Holim A, the homotopy limit of A, as Hom(I/, A). Then NHolim A is naturally isomorphic to holim NA (where for X: I S, holim X is defined in [1] as the simplicial set Hom(NI/, X)). A map φ: A B in CAT I induces a functor Holim φ from Holim A to Holim B. In fact, Holim is a functor from CAT I to CAT. Let L: J I be a functor. There is an obvious map in CAT J If S: I K in CAT, it is clear that L : J/ I/L (2.1) (SL) = S L L (2.2) Given a diagram J L I A CAT, we define a functor L : Holim A Holim (AL) as follows: On objects it is the composition CAT I (I/, A) L CAT J (I/L, AL) (L ) CAT J (J/, AL) where L is as in (2.1). Thus for α: I/ A, L α = J/ L I/L α L AL. The definition of L on morphisms is similar. It follows from (2.2) that (SL) = L S (2.3) We now describe the functorial behavior of Holim in a more organized manner. For any category C (not necessarily small) we define a category Dir(C) (which is never small): The objects are all functors I A C with I small. The maps from J B C to I A C are pairs (J L I, B φ AL). The composition law is: (J L I, B φ AL) (K S J, C ψ BS) = (K LS I, C φ S ψ ALS) We also need the category Inv(C) which we now define: It has the same objects as Dir(C). The maps from I A C to J B C are pairs (J L I, AL φ B). The composition law is: (K S J, BS ψ C) (J L I, AL φ B) = (K LS I, ALS ψ φ S C) Thus Inv(C) is isomorphic to Dir(C op ) op. We show that Holim is a functor from Inv(CAT ) to CAT : Given a map in Inv(CAT ), say (J L I, AL φ B): (I A CAT ) (J B CAT ) we define (L, φ) : Holim A Holim B as the composition (Holim φ) L. Note that if (K S J, BS ψ C): (J B CAT ) (K C CAT ) is another map in Inv(CAT ), then ((S, ψ) (L, φ)) = (LS, ψφ S ) = Holim (ψφ S ) (LS) = Holim ψ Holim φ S S L, and (S, ψ) (L, φ) = Holim ψ S Holim φ L and these two are equal, since Holim φ S S = S Holim φ. Thus Holim is indeed a functor from Inv(CAT ) to CAT. 2

3 3. A COFINALITY THEOREM We now show that NL is a homotopy equivalence if L has a right adjoint. This would follow from the cofinality theorem XI.9.2 of [1] if all NA i were fibrant: L is left cofinal ([1], Def. XI.9.1) since for all objects i of I, the category L/i has a final object. Notice that the maps in Hom(A, B) from F to G (here A and B are functors from I to CAT, and F and G are maps from A to B) may be identified with families {φ i : F i G i in B(i) A(i) } i ob I (thus for i ob I and a ob A(i), φ i;a is a map in B(i) from F i (a) to G i (a)) such that for all γ: i i in I and all objects a of A(i) B(γ)(φ i;a ) = φ i ;A(γ)(a) (3.4) Also notice that a map φ: F G of functors C C and a map ψ: H K of functors C C yield a commutative square of maps of functors C C : HF ψ F KF Hφ ψ G HG Kφ KG (3.5) Finally, notice that (3.5) generalizes to the case where F, G, H, and K are maps of functors I CAT : A map φ: F G in Hom(A, B) and a map ψ: H K in Hom(B, C) yield a commutative square in Hom(A, C) HF ψ ;F KF Hφ ψ Kφ ;G HG KG (3.6) where (ψ ;F ) i;a = ψ i;fi(a) and (Hφ) i;a = H i (φ i;a ) for i ob I and a ob A(i). Recall from (2.1) the map L : J/ I/L. Fix a functor M: J I and a map φ: L M. There is then a map in Hom(J/, I/M): φ : I/φ L M (3.7) Given an object j of J and an object j 0 j of J/j the map φ j;j0 j in I/Mj from J/φ j L j (j 0 j) = Lj 0 Lj φj M j to M j (j 0 j) = Mj 0 Mj is given by φ j0 Lj 0 Mj0. We derive some properties of φ : Given a functor S: I K, we have Sφ: SL SM and, by (3.7), (Sφ) : K/Sφ (SL) (SM), i. e., by (2.2), (Sφ) : K/Sφ S L L S M M, i. e., by (3.5), (Sφ) : S M I/φ L S M M. In fact, Further, if S: K J then φ S : LS MS and, by (3.7), (Sφ) = S M φ (3.8) (φ S ) : I/φ S (LS) (MS), i. e., by (2.2), (φ S ) : I/φ S L S S M S S. In fact, Finally, if S: J I and ψ: M S then ψφ: L S and (φ S ) = φ S;S (3.9) 3

4 (ψφ) : I/(ψφ) L S. In fact, (ψφ) = ψ (I/ψ)φ (3.10) 3.1. Theorem: Let L: J I and A: I CAT be functors, the former between small categories. If L has a right adjoint then L : Holim A Holim (AL) has a left adjoint Corollary: Let L: J I and A: I CAT be functors, the former between small categories. If L has a right adjoint then NL : NHolim A NHolim (AL) is a homotopy equivalence. Proof. Given a diagram j λ i ρ j of small categories and functors, there is a bijection between isomorphisms κ: i(λ.,.) j(., ρ.) and pairs (a: λρ 1 i, b: 1 j ρλ) satisfying a λ λb = 1 λ and ρa b ρ = 1 ρ. See 6 of [2] for the definition of the map κ (a, b). That this map is a bijection is clear. Fix a right adjoint R of L, and maps a: LR 1 I and b: 1 J RL as above. Let R = Holim Aa R : Holim (AL) Holim A. We show that R is left adjoint to L. Recall the map a : I/a (LR) 1 of (3.7). For any object α of Holim A, the following diagram shows that a α:= αa is a map in Holim A from R L (α) to α: I/ I/a α I/LR (ii) α LR (LR) (i) ALR Aa R L (α) I/ A Above, (i) commutes by the definitions of L and R and by (2.2), and (ii) commutes by (3.5). We show that a is natural: Fix φ: α α in Holim A. Apply (3.6) to a : I/a(LR) 1 in Hom(I/, I/) and φ: α α in Hom(I/, A) to conclude that: α a φ ;I/a(LR) = φ αa, i. e., a α φ ;I/a(LR) = φ a α (3.11) Now R L (φ) = (Aa)φ LR;(LR) and (3.4) implies that (Aa)φ LR; = φ ;I/a. Thus (3.11) shows that a is natural. Now recall the map b : J/b (RL) of (3.7). For any object β of Holim (AL), the following diagram shows that b β := (Aa L β RL )b is a map in Holim (AL) from β to L R (β): 4

5 J/ (RL) L R (β) (i) J/RL J/b β RL (ii) ALb ALRL (iii) β J/ AL Aa L AL 1 Above, (i) commutes by the definiitons of L and R and by (2.2), (ii) commutes by (3.5), and (iii) commutes since a L Lb = 1 L. We show that b is natural: Fix ψ: β β in Holim (AL). Apply (3.6) to the maps b : J/b (RL) in Hom(J/, J/RL) and ψ RL; : β RL β RL in Hom(J/RL, ALRL) to conclude that: β RLb ψ RL;J/b = ψ RL;(RL) β RL b, therefore (Aa L )β RLb (Aa L )ψ RL;J/b = (Aa L )ψ RL;(RL) (Aa L )β RL b, i. e., b β (Aa L)ψ RL;J/b = L R (ψ) b β (3.12) Now Aa L ALb = 1 and (3.4) implies that (ALb)ψ = ψ RL;J/b. Thus (3.12) shows that b is natural. We now show that a R R b = 1: Fix β ob Holim (AL). Then: a R β = (R β)a = (Aa β R R )a R b β = R ((Aa L β RL )b ) = (Aa Aa LR β RLR )b R;R But (3.5) implies that Aa LR β RLR = β R J/Ra, thus: a R β R b β = (Aa β R)(R a (J/Ra)b R;R ) = (by (3.8) and (3.9))(Aa β R )((Ra) J/Ra(b R ) ) = (by 3.10)) (Aa β R )(Ra b R ) = (Aa β R )1 R = (since R β = (Aa β R R ) 1 R β We conclude the proof by showing that L a b L = 1 L. Fix α ob Holim A. Then: L a α = L (αa ) = α L a L;L b L α = (Aa L (L α) RL )b = (Aa L (α L L ) RL )b = (Aa L α LRL L RL )b = (by (3.5)) (α L I/a L L RL )b, therefore L a α b L α = α L(a L;L (I/a L L RL )b ) = (by (3.8) and (3.9)) α L ((a L ) I/a L (Lb) ) = (by (3.10)) α L (a L Lb) = α L 1 L = (since L α = α L L ) 1 L α. 5

6 4. HOMOTOPY LIMITS OF DIAGRAMS OF CATEGORIES We now derive some naturality properties of Holim which will be needed in [4]: We will show that there is a functor Holim : Inv(CAT D ) CAT D, where D is any small category. For any category C, we define an embedding as follows: Write the usual isomorphism (C D ) I Inv(C D ) Inv(C) D (4.13) (C I ) D as (A φ B) (Â ˆφ ˆB). Explicitly, Â(d)(i) = A(i)(d) and ˆφ di = φ id. The embedding of (4.13) takes an object I A C D of Inv(C D ) to the object D Ǎ Inv(C) of Inv(C) D, where Ǎ = D Â C I Inv(C). It takes a map (J L I, AL φ B): (I A C D ) (J B C D ) of Inv(C D ) to the map (L, φ) : Ǎ ˇB of Inv(C) D, where (L, φ) d is given by (L, ˆφ d ): Ǎ(d) ˇB(d). Note Ǎ(d) = Â(d): I C and ˆφ d : (AL) (d) = Â(d)L ˆB(d), i. e., (L, φ) d is indeed a map from Ǎ(d) to ˇB(d) in Inv(C). We define the functor Holim : Inv(CAT D ) CAT D as the composition Inv(CAT D ) Inv(CAT ) D Holim CAT D where the first map is the map of (4.13). Note that by definition (Holim A)(d) is just Holim (Â(d)), and (L, φ) d = (L, ˆφ d ). We end this paper by describing a certain variation of Holim, which we call holim, and which is the one that is used in [4]. It is the composition Inv(op) Inv(CAT ) Holim Inv(CAT ) op CAT CAT Above, Inv(op) takes I A CAT to I A CAT op CAT and it takes (L, φ) to (L, op φ) Theorem: Let L: J I and A: I CAT be functors, the former between small categories. If L has a right adjoint then so does L : holim A holim (AL) Corollary: Let L: J I and A: I CAT be functors, the former between small categories. If L has a right adjoint then NL : Nholim A Nholim (AL) is a homotopy equivalence. Proof. The map L (resp. R ) in the proof of Thm. 3.1 is equal to (L, 1) = Holim (L, 1) (resp. (R, Aa) = Holim (R, Aa)). It follows that holim (L, 1) is now left adjoint to holim (R, Aa). Finally, the functor holim : Inv(CAT D ) CAT D is defined as the composition Inv(CAT D ) Inv(CAT ) D holim where the first map is the one in (4.13). CAT D Acknowledgements The author is grateful to the SFB 343 at the University of Bielefeld, especially Friedhelm Waldhausen, for their hospitality while this paper was written. References [1] A. K. Bousfield, D. M. Kan, Homotopy limits, localizations, and completions, Springer Lecture Notes Notes 304, [2] D. M. Kan, Adjoint functors, Trans. A. M. S. 87 (1958), [3] M. G. Lydakis, thesis, Brown University (1993). [4] M. G. Lydakis, Fixed point problems, equivariant stable homotopy, and a trace map for the algebraic K-theory of a point, to appear in Topology. 6

7 [5] G. B. Segal, Classifying spaces and spectral sequences, Publ. Math. I. H. E. S. 34 (1968), [6] R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Camb. Phil. Soc. 85 (1979), Fakultät für Mathematik Universität Bielefeld Bielefeld Germany 7

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in path component sheaves, and the diagrams Cocycle categories Cocycles J.F. Jardine I will be using the injective model structure on the category s Pre(C) of simplicial presheaves on a small Grothendieck site C. You can think in terms of simplicial