THE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS

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1 THE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS FINNUR LÁRUSSON Abstract. We give a detailed exposition o the homotopy theory o equivalence relations, perhaps the simplest nontrivial example o a model structure. Contents 1. Introduction 1 2. The model structure 2 3. Good properties o the model structure 5 4. Eective monomorphisms 8 5. Homotopy limits and colimits 9 Reerences Introduction Abstract homotopy theory, also known as homotopical algebra, is an abstraction o the homotopy theory o topological spaces. Its undamental concept is the notion o a model category, or a model structure on a category, introduced by Quillen [11] in Model structures have since appeared and been applied in a growing variety o mathematical areas. Abstract homotopy theory is a rather technical subject. It is usually quite diicult to veriy the deining properties o a model category. This note is a detailed treatment o what I believe to be the simplest nontrivial example o a model category: the category o equivalence relations. As ar as I know, such a treatment has not appeared in the literature beore. Equivalence relations have a role to play in applied homotopy theory in situations where two objects are either equivalent or not and that is all there is to it. Sometimes there is no natural notion o objects being equivalent in more than one way or o higher equivalences; sometimes one may simply wish to ignore such additional eatures. There are applications in my own work where more sophisticated structures such as groupoids and simplicial sets are unnecessarily complicated and using equivalence relations is the natural way to go. Such applications will typically involve embedding a geometric category in a category o diagrams or sheaves o equivalence relations. The purpose o this note is to provide a detailed account o the basics as a oundation or applications. Also, students o abstract homotopy theory, reading, say, [5], [6], or [7], may ind it useul to see the central ideas o homotopical algebra presented in a concrete example with much lighter technical baggage than the standard example, simplicial sets, requires. We start by deining our model structure on the category o equivalence relations, adopting the standard deinition or groupoids, and veriying Quillen s axioms. The homotopy category turns out to be the category o sets: the subject is homotopy theory at the level o π 0 and yet it is not trivial. We prove that the model structure is proper and combinatorial, but not cellular, unlike the usual model structures on simplicial sets and topological spaces. There Date: 30 October Minor changes 22 April Mathematics Subject Classiication. Primary 18G55. Secondary 55U35. Key words and phrases. Category, equivalence relation, partition, partitioned set, model structure. 1

2 is an enrichment o the category in itsel, which interacts well with the model structure, as shown by a version o Quillen s Axiom SM7. We introduce the pointwise ibration structure on the category o presheaves o equivalence relations over a small category and derive its basic properties. Finally, we prove that colimits o diagrams o equivalence relations preserve acyclic maps, so homotopy colimits are just ordinary colimits, point out that limits do not, and give an explicit construction and a discussion o homotopy limits. We have tried to make the paper as sel-contained as possible. For deinitions and other background not provided in detail, we reer the reader to [6] and [7]. We have proved those o our results that are elementary and most o them are using only basic set theory and category theory: or aesthetic reasons, to make the proos more accessible, and to lay bare the elementary nature o these results. We have avoided unnecessarily sophisticated machinery that in some cases would have yielded shorter proos. The main exceptions are results about categories o diagrams in E rather than E itsel: or these, deeper category theory and homotopy theory is needed, and precise reerences are provided. 2. The model structure An equivalence relation on a set is most simply viewed as a partition o the set. A partitioned set can be considered both as a topological space that is a disjoint union, with the coproduct topology, o nonempty spaces with a trivial topology, and as a groupoid in which there is at most one isomorphism rom one object to another. We usually denote by X or simply the equivalence relation corresponding to a partition o a set X. The quotient set X/ is sometimes denoted X. A morphism : X Y o partitioned sets is a map o sets that takes equivalent elements to equivalent elements. In other words, the image by o a class in X is contained in a class in Y. I X and Y are viewed as topological spaces, this is precisely the deinition o a continuous map; i they are viewed as groupoids, this is precisely the deinition o a morphism o groupoids. The induced map X Y is denoted. The category E o partitioned sets (more precisely, the category in which an object is a partition o a set and an arrow is a morphism o partitioned sets) is a ull subcategory o the category o topological spaces and o the category o groupoids. To provide a proper set-theoretic oundation or our study, we assume that a Grothendieck universe has been chosen and that all sets under consideration are elements o this universe (see [10], Sec. I.6). A category is called small i its objects orm a set. Deinition 1. A map : X Y o partitioned sets is said to be: (1) a coibration i is injective; (2) a ibration i maps each class o X onto a class o Y ; (3) a weak equivalence i induces a bijection o quotient sets. A weak equivalence is also called an equivalence or an acyclic map. These notions agree with the usual ones or groupoids, introduced by Anderson in [2]; or more details, see [8] or [12]. From the topological point o view, a ibration o partitioned sets as deined above is nothing but a Hurewicz ibration or equivalently a Serre ibration, and a weak equivalence is nothing but a topological weak equivalence or equivalently a homotopy equivalence. A coibration o partitioned sets, however, is more general than the two topological notions (both o which require a closed image, or example). Here are three important maps o partitioned sets: coibrations i 0 and i 1, and an acyclic coibration j. i 0 i 1 2 j

3 The ollowing result is immediate. It almost says that the model structure we are about to deine is coibrantly generated (an additional set-theoretic regularity property is needed: see Theorem 8). Proposition 2. (1) A map o partitioned sets is a ibration i and only i it has the right liting property with respect to j. (2) A map o partitioned sets is an acyclic ibration, that is, an acyclic surjection, i and only i it has the right liting property with respect to i 0 and i 1. The next theorem is the main result o this section. Theorem 3. The category E o partitioned sets with the three classes o maps deined above is a model category. This means the ollowing. (1) E has all small limits and colimits. (2) The two-out-o-three property: I and g are composable maps such that two o, g, and g are acyclic, then so is the third. (3) I is a retract o g, and g is acyclic, a coibration, or a ibration, then so is. (4) Every commuting square A X s j p B g Y where j is a coibration and p is a ibration and one o them is acyclic, has a liting s making the two triangles commute. (5) Every map can be unctorially actored into a coibration ollowed by an acyclic ibration, and into an acyclic coibration ollowed by a ibration. Proo. (1) The limit o a small diagram o partitioned sets is the set-limit with the coarsest partition (the largest equivalence relation) making the maps rom the limit to the sets in the diagram morphisms o partitioned sets. The colimit o a small diagram o partitioned sets is the set-colimit with the inest partition (the smallest equivalence relation) making the maps rom the sets in the diagram to the colimit morphisms o partitioned sets. (2) Consider the maps and ḡ o quotient sets induced by and g. Clearly, i two o, ḡ, and ḡ are bijections, then so is the third. (3) A retract o an injection is an injection. I is a retract o g, then is a retract o ḡ, and a retract o a bijection is a bijection. As or ibrations, we observe that retractions preserve right liting properties and invoke Proposition 2. (4) First, suppose j is acyclic and let b B. Since j is acyclic, there is a A with b j(a). Then g(b) g(j(a)) = p((a)), so since p is a ibration, g(b) = p(x) or some x (a). I b j(a), say b = j(a) with a A, we take x = (a). Set s(b) = x. Then p(s(b)) = p(x) = g(b), so s is a liting in the square. To veriy that s respects partitions, say b b in B. Find a, a A with j(a) b b j(a ). Since j is acyclic, a a, so s(b) (a) (a ) s(b ). Next, suppose p is acyclic, so p is surjective, and let b B. There is x X with g(b) = p(x). I b j(a), say b = j(a) with a A, we take x = (a). Set s(b) = x. Then p(s(b)) = p(x) = g(b), so s is a liting in the square. To veriy that s respects partitions, say b b in B. Then p(x) = g(b) g(b ) = p(x ) so s(b) = x x = s(b ) since p is acyclic. (5) We imitate the constructions o mapping cylinders and mapping path spaces in topology. Instead o the interval, we use the two-point set I = {0, 1} with 0 1. Let : X Y 3

4 be a map o partitioned sets. Consider the diagram X ι X I Y M p Y in E, where M is the pushout Y (X I), ι(x) = (x, 0), and (x, t) = (x). This gives a unctorial actorization = p j, where j : X M, x (x, 1). Clearly, j is injective, that is, a coibration. Also, p is surjective, so to veriy that p is an acyclic ibration, we need to show that p is injective. For y Y, p 1 (y) is the image in M o the subset {y} 1 (y) I o Y (X I). This image lies in a single class in M, since i x 1 (y), then y is identiied in M with (x, 0), which in turn is equivalent to (x, 1). Hence, i y y, then p 1 (y) and p 1 (y ) lie in a single class in M, so the preimage by p o the class o y is a single class in M. Next, we deine the path space Y I as the set o all maps I Y o partitioned sets, that is, maps taking both 0 and 1 to the same class in Y, with two such maps considered equivalent i the corresponding classes are the same. Note that this makes Y and Y I weakly equivalent. Consider the diagram X i P Y I e X Y where P is the pullback X Y I, e(α) = α(0), and (x) is the constant path taking both 0 and 1 to (x). This gives a unctorial actorization = q i, where q : P Y, (x, α) α(1). Clearly, i is injective. To see that i is acyclic, note that or x, x X, we have (x, (x)) = i(x) i(x ) = (x, (x )) i and only i x x ; also, i (x, α) P, then (x, α) (x, (x)) = i(x). Finally, to veriy that q is a ibration, take (x, α) P (so α(0) = (x)) and let y q(x, α) = α(1). Deine β : I Y, β(0) = (x), β(1) = y. Then β(0) = (x) = α(0) α(1) y = β(1), so β Y I, β α, and q(β) = y. The initial object o E is the empty set (the empty colimit) and the inal object o E is the one-point set (the empty limit), each with its unique partition. Clearly, or every partitioned set X, the canonical map X is a coibration and the canonical map X is a ibration, so X is both coibrant and ibrant, that is, biibrant. Thus, by the Whitehead Lemma (see [7], Thm ), every weak equivalence o partitioned sets is a homotopy equivalence. There is a unctor Q rom E to the category Set o sets, taking a partitioned set to its quotient set. This unctor has a right adjoint R : Set E, endowing a set with its discrete partition. Namely, or a set A and a partitioned set X, there is a natural bijection between maps QX A and morphisms X RA. (It is an exercise or the reader to show that Q has no let adjoint.) Now, Set has a rather trivial model structure in which the isomorphisms, that is, the bijections, are the weak equivalences and every map is both a coibration and a ibration. The pair (Q, R) is then a Quillen pair, meaning that Q preserves coibrations and R preserves ibrations (see [7], Sec. 8.5). Furthermore, (Q, R) is a pair o Quillen equivalences, meaning that the map QX A is acyclic i and only i the corresponding morphism X RA is acyclic. This implies that the homotopy categories o E and o Set are equivalent; the latter is clearly Set itsel, and we have proved the ollowing result. 4

5 Theorem 4. The homotopy category o E is equivalent to the category o sets. This does not mean that the homotopy theory o equivalence relations is trivial: there is more to a model structure than its homotopy category. While a model structure is usually viewed as a tool or the study o the associated homotopy category, there are applications in which the model structure itsel (in particular the ibrations and coibrations) is the primary object o interest. An example is the study [9] o liting and extension properties in complex analysis. Moreover, applications o the theory presented here will likely involve localizations o the pointwise ibration structure on categories o diagrams in E (see Corollary 9). The homotopy categories o such localizations will typically be quite intricate. 3. Good properties o the model structure The irst three theorems in this section show that the model structure on E has many o the good properties that we like model categories to have. First, a model structure is said to be let proper i the pushout o an acyclic map along a coibration is acyclic, right proper i the pullback o an acyclic map along a ibration is acyclic, and proper i it is both let proper and right proper. Theorem 5. The model category E is proper. Proo. Consider a commuting square A h B C k D in E. First assume the square is a pushout, h is a coibration, and is acyclic. We need to show that g is acyclic. We can take D = (B C)/, where is the equivalence relation generated by letting (a) h(a) or every a A. The equivalence relation making D a partitioned set is then generated by letting g(b) g(b ) whenever b b in B, and k(c) k(c ) whenever c c in C. To show that ḡ is injective, suppose b, b B and g(b) g(b ). By the descriptions o and just given, there is an even number m 0 and points b 1,..., b m B and c 1,..., c m C such that b b 1 c 1 c 2 b 2 b 3 c 3 c m b m b. Also, b j c j means that there is an odd number i j 1 and points a 1 j,..., a i j j A such that b j = h(a 1 j), (a 1 j) = (a 2 j), h(a 2 j) = h(a 3 j), (a 3 j) = (a 4 j),..., h(a i j 1 j ) = h(a i j j ), and c j = (a i j j ). Since h is injective, a2 j = a 3 j, a 4 j = a 5 j,..., a i j 1 j (a i j j ) = c j. Thus, writing a j = a 1 j, we now have g = a i j j, so (a1 j) = = b h(a 1 ) (a 1 ) (a 2 ) h(a 2 ) h(a 3 ) (a 3 ) (a m ) h(a m ) b. Since is acyclic, we get a 1 a 2, a 3 a 4,..., a m 1 a m, so b h(a 1 ) h(a 2 ) h(a 3 ) b and b b. I d D and d / g(b), then d = k(c) or some c C. Since is acyclic, c (a) or some a A. Then g(h(a)) = k((a)) k(c) = d. This shows that ḡ is surjective. Right properness is easier. Assume the square is a pullback, k is a ibration, and g is acyclic. We need to show that is acyclic. We can take A = {(b, c) B C : g(b) = k(c)} with (b, c) (b, c ) i and only i b b and c c. I (b, c), (b, c ) A and c = (b, c) 5

6 (b, c ) = c, then g(b) = k(c) k(c ) = g(b ), so b b since g is acyclic, and (b, c) (b, c ). Also, i c C, there is b B with g(b) k(c) since g is acyclic. Since k is a ibration, there is c c with k(c ) = g(b); then (b, c ) A and (b, c ) = c c. I X and Y are partitioned sets, the set hom(x, Y ) o morphisms X Y carries an equivalence relation such that g i = ḡ, that is, i (x) g(x) or all x X. We write Hom(X, Y ) or the set hom(x, Y ) with this equivalence relation. Every composition map in particular the evaluation map Hom(X, Y ) Hom(Y, Z) Hom(X, Z), (, g) g, e : X Hom(X, Y ) Y, (x, ) (x), is a morphism o partitioned sets. This enrichment o the category E in itsel interacts well with the model structure on E. More precisely, we have the ollowing version o Quillen s Axiom SM7. Theorem 6. I j : A B is a coibration and p : X Y is a ibration o partitioned sets, then the induced map (j, p ) : Hom(B, X) Hom(A, X) Hom(A,Y ) Hom(B, Y ) is a ibration o partitioned sets, which is acyclic i j or p is acyclic. We ollow the usual method o proo or simplicial sets, as in [6], Sec. I.5. The ollowing lemma is called the Exponential Law. Lemma 7. For partitioned sets A, X, and Y, the map e : Hom(A, Hom(X, Y )) Hom(X A, Y ), e (g)(x, a) = g(a)(x), is an isomorphism o partitioned sets, which is natural in A, X, and Y. Proo. The inverse morphism satisies e 1 (h)(a)(x) = h(a, x). Proo o Theorem 6. Let i : K L be a coibration o partitioned sets. By the Exponential Law, there is a liting in a square o the orm K Hom(B, X) i (j,p ) L Hom(A, X) Hom(A,Y ) Hom(B, Y ) i and only i there is a liting in the corresponding square (K B) (K A) (L A) X ι L B Clearly, ι is injective and thus a coibration. To conclude the proo, we need to show that ι is acyclic i either i or j is. Say i is (the other case is analogous). Then i id B is acyclic and, by Theorem 5, so is (id K j) (i id A ). The ormer map is ι precomposed by the latter map, so ι is acyclic by the two-out-o-three property. Theorem 8. The model category E is locally initely presentable and coibrantly generated, and hence combinatorial. 6 Y p

7 Proo. Note that, irst, every partitioned set is the colimit o the directed diagram o all its inite subsets, ordered by inclusion; second, a partitioned set X is initely presentable, meaning that hom(x, ) preserves directed colimits in E, i and only i it is inite; and, third, there is, up to isomorphism, only a set o inite partitioned sets. Since E is cocomplete, this shows that E is locally initely presentable (see [1], Ch. 1). The domains o the maps i 0, i 1, and j in Proposition 2 are inite and hence initely presentable in E, that is, ℵ 0 -small relative to E in the language o [7]. Thus, E is coibrantly generated with generating coibrations i 0 and i 1 and a generating acyclic coibration j (see [7], De ). Finally, a locally presentable coibrantly generated model category is, by deinition, combinatorial. Every object in a locally presentable category is in act presentable (small in the language o [7]; see [1], Prop. 1.16). Hence, a locally presentable model category is coibrantly generated, and thus combinatorial, i and only i its ibrations are characterized by a right liting property with respect to a set o acyclic coibrations and its acyclic ibrations are characterized by a right liting property with respect to a set o coibrations. A good reerence or basic acts on locally presentable model categories is [3], Sec. 1. An important such act is that the so-called small object argument works or any set o morphisms in a locally presentable category. See also [4], Sec. 2. Let us say a ew more words about the category theory o E. Let F be the small, ull subcategory o E o inite partitioned sets. The canonical unctor rom E to the category Set F op o presheaves o sets on F is a ull embedding, preserves directed colimits, and has a let adjoint, so it preserves limits (see [1], Prop s 1.26, 1.27). Hence, E is equivalent to a ull, relective subcategory o the topos Set F op, closed under directed colimits. However, E itsel is not a topos, i only because it lacks a subobject classiier. We conclude this section with two results about the categories that one would actually use in geometric applications o the homotopy theory o equivalence relations. The irst result introduces the so-called pointwise ibration structure on categories o presheaves o equivalence relations. A preshea on a category C thought o as a site is the same thing as a diagram over the opposite category C op thought o as an indexing category. The two points o view are equivalent, but the ormer is more common in geometric applications. Corollary 9. Let C be a small category and consider the category E Cop o presheaves o equivalence relations on C. There is a model structure on E Cop in which weak equivalences and ibrations are deined pointwise and coibrations are deined by the let liting property with respect to acyclic ibrations. This model structure is proper and coibrantly generated. It is also locally initely presentable and hence combinatorial. Proo. The existence o the speciied model structure and it being proper and coibrantly generated ollows rom our previous theorems and the results o [7], Sec s 11.6, By [1], Cor. 1.54, E Cop is locally initely presentable since E is. The coibrations in the pointwise ibration structure can be described somewhat explicitly: see [7], Thm The category E Cop is enriched in E in much the same way that E itsel is. Namely, i X and Y are E-valued presheaves on C, the set hom(x, Y ) o morphisms X Y carries an equivalence relation such that φ ψ i φ C and ψ C are equivalent as maps X(C) Y (C) o partitioned sets or every object C in C. We write Hom(X, Y ) or the set hom(x, Y ) with this equivalence relation. As beore, every composition map Hom(X, Y ) Hom(Y, Z) Hom(X, Z), is a morphism o partitioned sets, and we have a version o Quillen s Axiom SM7. 7

8 Theorem 10. I j : A B is a coibration and p : X Y is a ibration in E Cop, then the induced map (j, p ) : Hom(B, X) Hom(A, X) Hom(A,Y ) Hom(B, Y ) is a ibration in E, which is acyclic i j or p is acyclic. To prove this, we start with a variant o the Exponential Law. I K is a partitioned set and X is an object in E Cop, we deine the object X K in E Cop by setting X K (C) = Hom(K, X(C)) and letting a map ρ : C D in C induce a map Hom(K, X(D)) Hom(K, X(C)) by postcomposition by ρ : X(D) X(C). This construction is easily veriied to be covariant in X and contravariant in K. Lemma 11. For E-valued presheaves X and Y on C and a partitioned set K, the map ɛ : Hom(K, Hom(X, Y )) Hom(X, Y K ), ɛ(g) C (x)(k) = g(k) C (x), or each object C o C, x X(C), and k K, is an isomorphism o partitioned sets, which is natural in X, Y, and K. Proo. The inverse morphism satisies ɛ 1 (h)(k) C (x) = h C (x)(k). Proo o Theorem 10. Let i : K L be a coibration o partitioned sets. By the Exponential Law, there is a liting in a square o the orm K i Hom(B, X) (j,p ) L Hom(A, X) Hom(A,Y ) Hom(B, Y ) i and only i there is a liting in the corresponding square j A B X L q X K Y K Y L We need to veriy that q is a ibration which is acyclic i either i or p is. Since the ibrations and weak equivalences in E Cop are deined pointwise, this ollows directly rom Theorem Eective monomorphisms Cellularity is an important strengthening o coibrant generation (see [7], Ch. 12). The usual model structures on the category o simplicial sets and the category o topological spaces (say compactly generated and weakly Hausdor) are both cellular. We will show that E is not cellular. (For another example o a coibrantly generated model category that is not cellular, see [7], Ex ) Let proper cellular model categories admit let Bousield localization with respect to any set o maps. Fortunately, let proper combinatorial model categories do as well. Proposition 12. E is not cellular. Proo. By the deinition o cellularity, coibrations in a cellular model category are eective monomorphisms (some say regular monomorphisms), that is, equalizers o pairs o maps (see [7], De ). This ails in E. For instance, the coibration i 1 in Proposition 2 is not an equalizer. It is an easy exercise to show this directly. It also ollows rom the act that an eective monomorphism that is also an epimorphism is an isomorphism, whereas i 1 is both a monomorphism and an epimorphism, but not an isomorphism (the existence o such a map is another reason why E is not a topos). 8

9 This result prompts us to take a closer look at eective monomorphisms in E. Note that the monomorphisms in E are precisely the injections, that is, the coibrations. Proposition 13. A monomorphism m : A B in E is eective i and only i it induces an injection o quotient sets, that is, m(a) m(a ) in B i and only i a a in A. Proo. Since m is an injection, it is the set-limit o the diagram B B A B with the two natural inclusions, and m is the E-limit o this diagram i and only i A carries the largest equivalence relation making A B a morphism o partitioned sets (see the description o limits in E in the proo o Theorem 3). This last condition is also implied by m being the equalizer o any diagram B C, again by the description o limits in E. It is now natural to ask the ollowing question. Is there a model structure on E (cellular, one would hope) in which the coibrations are the eective monomorphisms and the weak equivalences are the same as beore? The answer is no: the acyclic coibrations would still be the same (injections that induce bijections o quotient sets), so the whole model structure would be the same. 5. Homotopy limits and colimits Very simple examples, such as the pullback square in E, where and g have dierent images, or the map o equalizers g c id id where c is constant, show that limits o diagrams o partitioned sets need not preserve acyclicity: the map o limits induced by a pointwise acyclic map o diagrams need not be acyclic. This orces us to study homotopy limits o diagrams o partitioned sets. Indeed, one important aspect o model structures in general is that we can use the coibrations and ibrations to construct and understand a modiication o the ordinary notion o limits that does respect weak equivalences. As or colimits, it is a special eature o the category o partitioned sets that they do preserve acyclicity, so homotopy colimits are just ordinary colimits. Theorem 14. The morphism o colimits induced by a pointwise acyclic map o diagrams o partitioned sets is acyclic. Proo. Every colimit is a coequalizer o a map o coproducts (see [10], Sec. V.2). Clearly, any set o weak equivalences o partitioned sets induces a weak equivalence rom the coproduct o the sources to the coproduct o the targets. Thus, we need to show that i we have a map o coequalizers o partitioned sets A X g α β h B Y k such that hα = β, kα = βg, and α and β are acyclic, then the induced map γ is also acyclic. We can take M = X/, where is the smallest equivalence relation on X with (a) g(a) 9 p q M γ N

10 or all a A. The equivalence relation on M is the smallest one making p a morphism, that is, the smallest one with p(x) p(x ) i x x in X. Analogous remarks hold or N. It is easy to show that γ is surjective: i n N, say n = q(y), then, since β is surjective, there is x X with y β(x), so n = q(y) qβ(x) = γp(x). To show that γ is injective, let m, m M with γ(m) γ(m ) in N. Say m = p(x), m = p(x ), so qβ(x) qβ(x ) in N. Write y = β(x), y = β(x ). The assumption that q(y) q(y ) in N means that there is a string y y 1 y 1 y 2 y ν y in Y. The question is whether we can lit this string to a string joining x and x in X. Since β is injective, this is clear i ν = 0. Assume ν 1 and consider the pair y 1 y 1. There are b 1,..., b j B and y 1 = z 0, z 1,..., z j = y 1 Y such that {z i 1, z i } = {h(b i ), k(b i )} or i = 1,..., j. One o the mutually analogous cases is when y 1 = h(b 1 ), k(b 1 ) = h(b 2 ),..., k(b j ) = y 1. Since ᾱ is surjective, there is a i A with b i α(a i ) or i = 1,..., j. Then h(b i ) hα(a i ) = β(a i ) and k(b i ) kα(a i ) = βg(a i ), so β(x) = y y 1 β(a 1 ), βg(a 1 ) k(b 1 ) = h(b 2 ) β(a 2 ),..., βg(a j ) k(b j ) = y 1. Since β is injective, we get x (a 1 ), g(a 1 ) (a 2 ),..., g(a j 1 ) (a j ). I ν = 1, then βg(a j ) y 1 y = β(x ), so g(a j ) x and we have a string joining x and x, showing that m m. I ν 2, we next consider the pair y 2 y 2 and get y 2 β(ã 1 ), say, so βg(a j ) y 1 y 2 β(ã 1 ) and g(a j ) (ã 1 ), thus continuing the string that will eventually join x and x. Note that we did not need injectivity o ᾱ in order to prove injectivity o γ. Indeed, i A, choose s A, adjoin a new element t to A such that t a or all a A \ {t}, and let, g, and α take s and t to the same points in their respective targets. Then the diagram is still a map o coequalizers, β and γ are still acyclic, but ᾱ is not injective any more. Now we turn to homotopy limits. Every limit is an equalizer o a map o products (see [10], Sec. V.2). It is easy to veriy that any set o weak equivalences o partitioned sets induces a weak equivalence rom the product o the sources to the product o the targets. We can thereore restrict our attention to equalizers. The general theory o homotopy limits is quite involved and is developed in detail in [7], Ch s 18, 19. See also [5], Sec. 10. Our deinition o homotopy equalizers in E is motivated by the general theory and justiied by the results that ollow. Deinition 15. The homotopy equalizer o a diagram o partitioned sets A g X is the set {a A : (a) g(a)} with the equivalence relation induced rom A. Let D be the indexing category. Then the unctor category E D is the category o diagrams A X in E and maps between them. Theorem 16. (1) The homotopy equalizer as deined above gives a unctor E D E with a natural transormation to the projection unctor that takes A X to A. (2) The homotopy equalizer unctor takes a pointwise acyclic map to an acyclic map. (3) Let be a diagram in E. The inclusion A g X {a A : (a) = g(a)} {a A : (a) g(a)} o the equalizer into the homotopy equalizer is acyclic i and only i or every a A with (a) g(a), there is a A with (a ) = g(a ) such that a a. 10

11 Proo. (1) It is easily veriied that a map in E D rom A X to B Y yields a commuting diagram H A X K B Y in E, where we have denoted the homotopy equalizers o A X and B Y by H and K respectively, and H A and K B are the inclusions. (2) We need to show that i we have a map o homotopy equalizer diagrams γ H A α g h β X K B Y k such that hα = β, kα = βg, and α and β are acyclic, then the induced map γ, obtained by restricting α, is also acyclic. Clearly, γ is injective since ᾱ is. To show that γ is surjective (this is what generally ails or ordinary equalizers), take b B with h(b) k(b) and, using surjectivity o ᾱ, ind a A such that α(a) b. Then β(a) = hα(a) h(b) k(b) kα(a) = βg(a), so (a) g(a) since β is injective. This shows that every element o K is equivalent to an element in the image o γ. (3) The inclusion induces an injection o quotient sets by the deinition o the equivalence relations on the equalizer and the homotopy equalizer: both are induced rom A. The given condition is precisely what it means or the inclusion to induce a surjection o quotient sets. There is a model structure on the diagram category E D in which the coibrations and the weak equivalences are deined pointwise. We call it the pointwise coibration structure. The relevance o such structures to homotopy limits is discussed in [5], Sec. 10. (The pointwise ibration structure described in Corollary 9 is similarly relevant to homotopy colimits.) It is easy to veriy that the characterization in Theorem 16 o when the homotopy equalizer o a diagram A X is weakly equivalent to its ordinary equalizer means precisely that A X (or rather the map rom A X to the inal object in E D ) has the right liting property with respect to the map α g h k 1 2 β where (1) = 1, g(1) = 2, α(1) = 1, β(1) = 1, β(2) = 2, h(1) = 1, h(2) = 3, k(1) = 2, k(2) = 3, and the sources and targets have only one equivalence class each. This map is an acyclic coibration in E D with the pointwise coibration structure, so the ollowing corollary is immediate. Corollary 17. I a diagram A X is ibrant in the pointwise coibration structure on E D, then the natural map rom its equalizer to its homotopy equalizer is acyclic. Dually, or every small category C and every coibrantly generated model category M, the map o colimits induced by a pointwise acyclic map o diagrams in M C is acyclic i the diagrams are coibrant in the pointwise ibration structure (see [7], Thm ). This does not imply our Theorem 14 because it is generally ar rom true that every diagram in E C 11

12 is coibrant in the pointwise ibration structure. For example, with C = D as above, the diagram, where the two arrows have the same image, has no map to the diagram, where the two arrows have dierent images, even though the map rom the latter diagram to the inal object in E D is a pointwise acyclic ibration. Let us consider the special case o homotopy pullbacks. The pullback o a diagram is the equalizer o the diagram A g C B F A B C, G where F (a, b) = (a) and G(a, b) = g(b). One usually takes the homotopy pullback by replacing one o the maps, say, with a ibration, meaning that one actors into an acyclic coibration A P ollowed by a ibration P C, and then taking the pullback B C P. Here, P can be taken to be the mapping path space A C I described in the proo o Theorem 3. This recipe or the homotopy pullback gives B C P = B C (A C I ) = {(b, a, α) B A C I : α(0) = (a), α(1) = g(b)}. Recalling that an element o C I, that is, a path in C, is simply a pair o equivalent elements o C, we see that this partitioned set is isomorphic to the set {(a, b) A B : (a) g(b)}, which is the homotopy equalizer o A B C as we have deined it. Finally, we remark that since limits and colimits in the preshea category E Cop are taken pointwise, homotopy limits in the pointwise ibration structure o Corollary 9 may be taken pointwise using Deinition 15. Also, by Theorem 14, a pointwise acyclic map o diagrams o presheaves induces an acyclic map o their colimits. Reerences [1] Adámek, J. and J. Rosický. Locally presentable and accessible categories. London Mathematical Society Lecture Note Series, vol Cambridge University Press, [2] Anderson, D. W. Fibrations and geometric realizations. Bull. Amer. Math. Soc. 84 (1978) [3] Beke, T. Sheaiiable homotopy model categories. Math. Proc. Cambridge Philos. Soc. 129 (2000) [4] Dugger, D. Combinatorial model categories have presentations. Adv. Math. 164 (2001) [5] Dwyer, W. G. and J. Spaliński. Homotopy theories and model categories. Handbook o algebraic topology, pp North-Holland, [6] Goerss, P. G. and J. F. Jardine. Simplicial homotopy theory. Progress in Mathematics, vol Birkhäuser Verlag, [7] Hirschhorn, P. S. Model categories and their localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, [8] Hollander, S. A homotopy theory or stacks. Preprint, 2001, arxiv:math.at/ (unpublished). [9] Lárusson, F. Model structures and the Oka Principle. J. Pure Appl. Algebra 192 (2004) [10] Mac Lane, S. Categories or the working mathematician. Graduate Texts in Mathematics, vol. 5. Springer- Verlag, [11] Quillen, D. G. Homotopical algebra. Lecture Notes in Mathematics, vol. 43. Springer-Verlag, [12] Strickland, N. P. K(N)-local duality or inite groups and groupoids. Topology 39 (2000) School o Mathematical Sciences, University o Adelaide, Adelaide SA 5005, Australia. address: innur.larusson@adelaide.edu.au 12