THE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS
|
|
- Ginger Barker
- 5 years ago
- Views:
Transcription
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
DUALITY AND SMALL FUNCTORS
DUALITY AND SMALL FUNCTORS GEORG BIEDERMANN AND BORIS CHORNY Abstract. The homotopy theory o small unctors is a useul tool or studying various questions in homotopy theory. In this paper, we develop the
More informationMADE-TO-ORDER WEAK FACTORIZATION SYSTEMS
MADE-TO-ORDER WEAK FACTORIZATION SYSTEMS EMILY RIEHL The aim o this note is to briely summarize techniques or building weak actorization systems whose right class is characterized by a particular liting
More information1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationHSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS
HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS MICHAEL BARR Abstract. Given a triple T on a complete category C and a actorization system E /M on the category o algebras, we show there is a 1-1 correspondence
More informationCLASS NOTES MATH 527 (SPRING 2011) WEEK 6
CLASS NOTES MATH 527 (SPRING 2011) WEEK 6 BERTRAND GUILLOU 1. Mon, Feb. 21 Note that since we have C() = X A C (A) and the inclusion A C (A) at time 0 is a coibration, it ollows that the pushout map i
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationJoseph Muscat Categories. 1 December 2012
Joseph Muscat 2015 1 Categories joseph.muscat@um.edu.mt 1 December 2012 1 Objects and Morphisms category is a class o objects with morphisms : (a way o comparing/substituting/mapping/processing to ) such
More informationON THE COFIBRANT GENERATION OF MODEL CATEGORIES arxiv: v1 [math.at] 16 Jul 2009
ON THE COFIBRANT GENERATION OF MODEL CATEGORIES arxiv:0907.2726v1 [math.at] 16 Jul 2009 GEORGE RAPTIS Abstract. The paper studies the problem of the cofibrant generation of a model category. We prove that,
More informationLIMITS AND COLIMITS. m : M X. in a category G of structured sets of some sort call them gadgets the image subset
5 LIMITS ND COLIMITS In this chapter we irst briely discuss some topics namely subobjects and pullbacks relating to the deinitions that we already have. This is partly in order to see how these are used,
More informationDescent on the étale site Wouter Zomervrucht, October 14, 2014
Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and
More informationPART I. Abstract algebraic categories
PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationUniversity of Cape Town
The copyright o this thesis rests with the. No quotation rom it or inormation derived rom it is to be published without ull acknowledgement o the source. The thesis is to be used or private study or non-commercial
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationTHE SNAIL LEMMA ENRICO M. VITALE
THE SNIL LEMM ENRICO M. VITLE STRCT. The classical snake lemma produces a six terms exact sequence starting rom a commutative square with one o the edge being a regular epimorphism. We establish a new
More informationCHOW S LEMMA. Matthew Emerton
CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there
More informationA Grothendieck site is a small category C equipped with a Grothendieck topology T. A Grothendieck topology T consists of a collection of subfunctors
Contents 5 Grothendieck topologies 1 6 Exactness properties 10 7 Geometric morphisms 17 8 Points and Boolean localization 22 5 Grothendieck topologies A Grothendieck site is a small category C equipped
More informationLOOP SPACES IN MOTIVIC HOMOTOPY THEORY. A Dissertation MARVIN GLEN DECKER
LOOP SPACES IN MOTIVIC HOMOTOPY THEORY A Dissertation by MARVIN GLEN DECKER Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationTangent Categories. David M. Roberts, Urs Schreiber and Todd Trimble. September 5, 2007
Tangent Categories David M Roberts, Urs Schreiber and Todd Trimble September 5, 2007 Abstract For any n-category C we consider the sub-n-category T C C 2 o squares in C with pinned let boundary This resolves
More informationLecture 9: Sheaves. February 11, 2018
Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with
More informationGENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions
More informationA Peter May Picture Book, Part 1
A Peter May Picture Book, Part 1 Steve Balady Auust 17, 2007 This is the beinnin o a larer project, a notebook o sorts intended to clariy, elucidate, and/or illustrate the principal ideas in A Concise
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More informationON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS FOR MODEL CATEGORIES
ON THE CONSTRUCTION OF FUNCTORIAL FACTORIZATIONS FOR MODEL CATEGORIES TOBIAS BARTHEL AND EMIL RIEHL Abstract. We present general techniques or constructing unctorial actorizations appropriate or model
More informationSpan, Cospan, and Other Double Categories
ariv:1201.3789v1 [math.ct] 18 Jan 2012 Span, Cospan, and Other Double Categories Susan Nieield July 19, 2018 Abstract Given a double category D such that D 0 has pushouts, we characterize oplax/lax adjunctions
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationGENERAL ABSTRACT NONSENSE
GENERAL ABSTRACT NONSENSE MARCELLO DELGADO Abstract. In this paper, we seek to understand limits, a uniying notion that brings together the ideas o pullbacks, products, and equalizers. To do this, we will
More informationON KAN EXTENSION OF HOMOLOGY AND ADAMS COCOMPLETION
Bulletin o the Institute o Mathematics Academia Sinica (New Series) Vol 4 (2009), No 1, pp 47-66 ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION B AKRUR BEHERA AND RADHESHAM OTA Abstract Under a set
More informationarxiv: v1 [math.ct] 28 Oct 2017
BARELY LOCALLY PRESENTABLE CATEGORIES arxiv:1710.10476v1 [math.ct] 28 Oct 2017 L. POSITSELSKI AND J. ROSICKÝ Abstract. We introduce a new class of categories generalizing locally presentable ones. The
More informationsset(x, Y ) n = sset(x [n], Y ).
1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,
More informationCategory Theory. Travis Dirle. December 12, 2017
Category Theory 2 Category Theory Travis Dirle December 12, 2017 2 Contents 1 Categories 1 2 Construction on Categories 7 3 Universals and Limits 11 4 Adjoints 23 5 Limits 31 6 Generators and Projectives
More informationRepresentation Theory of Hopf Algebroids. Atsushi Yamaguchi
Representation Theory o H Algebroids Atsushi Yamaguchi Contents o this slide 1. Internal categories and H algebroids (7p) 2. Fibered category o modules (6p) 3. Representations o H algebroids (7p) 4. Restrictions
More informationClassification of effective GKM graphs with combinatorial type K 4
Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,
More informationON THE CONSTRUCTION OF LIMITS AND COLIMITS IN -CATEGORIES
ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES EMILY RIEHL ND DOMINIC VERITY bstract. In previous work, we introduce an axiomatic ramework within which to prove theorems about many varieties o
More information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
More informationMODEL STRUCTURES ON PRO-CATEGORIES
Homology, Homotopy and Applications, vol. 9(1), 2007, pp.367 398 MODEL STRUCTURES ON PRO-CATEGORIES HALVARD FAUSK and DANIEL C. ISAKSEN (communicated by J. Daniel Christensen) Abstract We introduce a notion
More informationCartesian Closed Topological Categories and Tensor Products
Cartesian Closed Topological Categories and Tensor Products Gavin J. Seal October 21, 2003 Abstract The projective tensor product in a category of topological R-modules (where R is a topological ring)
More informationGrothendieck construction for bicategories
Grothendieck construction or bicategories Igor Baković Rudjer Bošković Institute Abstract In this article, we give the generalization o the Grothendieck construction or pseudo unctors given in [5], which
More informationDerived Algebraic Geometry IX: Closed Immersions
Derived Algebraic Geometry I: Closed Immersions November 5, 2011 Contents 1 Unramified Pregeometries and Closed Immersions 4 2 Resolutions of T-Structures 7 3 The Proof of Proposition 1.0.10 14 4 Closed
More informationCATEGORIES. 1.1 Introduction
1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the
More informationAn Introduction to Topos Theory
An Introduction to Topos Theory Ryszard Paweł Kostecki Institute o Theoretical Physics, University o Warsaw, Hoża 69, 00-681 Warszawa, Poland email: ryszard.kostecki % uw.edu.pl June 26, 2011 Abstract
More informationReview of category theory
Review of category theory Proseminar on stable homotopy theory, University of Pittsburgh Friday 17 th January 2014 Friday 24 th January 2014 Clive Newstead Abstract This talk will be a review of the fundamentals
More informationElementary (ha-ha) Aspects of Topos Theory
Elementary (ha-ha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................
More informationFUNCTORS BETWEEN REEDY MODEL CATEGORIES OF DIAGRAMS
FUNCTORS BETWEEN REEDY MODEL CATEGORIES OF DIAGRAMS PHILIP S. HIRSCHHORN AND ISMAR VOLIĆ Abstract. If D is a Reedy category and M is a model category, the category M D of D-diagrams in M is a model category
More informationEXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1. Categories 1.1. Generalities. I ve tried to be as consistent as possible. In particular, throughout the text below, categories will be denoted by capital
More informationAmalgamable diagram shapes
Amalgamable diagram shapes Ruiyuan hen Abstract A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 2 RAVI VAKIL CONTENTS 1. Where we were 1 2. Yoneda s lemma 2 3. Limits and colimits 6 4. Adjoints 8 First, some bureaucratic details. We will move to 380-F for Monday
More informationENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM FOR ARTIN STACKS
ENHANCED SIX OPERATIONS AND BASE CHANGE THEOREM FOR ARTIN STACKS YIFENG LIU AND WEIZHE ZHENG Abstract. In this article, we develop a theory o Grothendieck s six operations or derived categories in étale
More informationCategory Theory (UMV/TK/07)
P. J. Šafárik University, Faculty of Science, Košice Project 2005/NP1-051 11230100466 Basic information Extent: 2 hrs lecture/1 hrs seminar per week. Assessment: Written tests during the semester, written
More informationThe basics of frame theory
First version released on 30 June 2006 This version released on 30 June 2006 The basics o rame theory Harold Simmons The University o Manchester hsimmons@ manchester.ac.uk This is the irst part o a series
More informationUMS 7/2/14. Nawaz John Sultani. July 12, Abstract
UMS 7/2/14 Nawaz John Sultani July 12, 2014 Notes or July, 2 2014 UMS lecture Abstract 1 Quick Review o Universals Deinition 1.1. I S : D C is a unctor and c an object o C, a universal arrow rom c to S
More informationON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES
ON THE HOMOTOPY THEORY OF ENRICHED CATEGORIES CLEMENS BERGER AND IEKE MOERDIJK Abstract. We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given
More informationGraduate algebraic K-theory seminar
Seminar notes Graduate algebraic K-theory seminar notes taken by JL University of Illinois at Chicago February 1, 2017 Contents 1 Model categories 2 1.1 Definitions...............................................
More information1. Introduction and preliminaries
Quasigroups and Related Systems 23 (2015), 283 295 The categories of actions of a dcpo-monoid on directed complete posets Mojgan Mahmoudi and Halimeh Moghbeli-Damaneh Abstract. In this paper, some categorical
More informationDerived Algebraic Geometry I: Stable -Categories
Derived Algebraic Geometry I: Stable -Categories October 8, 2009 Contents 1 Introduction 2 2 Stable -Categories 3 3 The Homotopy Category of a Stable -Category 6 4 Properties of Stable -Categories 12 5
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationStabilization as a CW approximation
Journal of Pure and Applied Algebra 140 (1999) 23 32 Stabilization as a CW approximation A.D. Elmendorf Department of Mathematics, Purdue University Calumet, Hammond, IN 46323, USA Communicated by E.M.
More informationA calculus of fractions for the homotopy category of a Brown cofibration category
A calculus o ractions or the homotopy category o a Brown coibration category Sebastian Thomas Dissertation August 2012 Rheinisch-Westälisch Technische Hochschule Aachen Lehrstuhl D ür Mathematik ii Version:
More informationA CHARACTERIZATION OF CENTRAL EXTENSIONS IN THE VARIETY OF QUANDLES VALÉRIAN EVEN, MARINO GRAN AND ANDREA MONTOLI
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 15 12 CHRCTERIZTION OF CENTRL EXTENSIONS IN THE VRIETY OF QUNDLES VLÉRIN EVEN, MRINO GRN ND NDRE MONTOLI bstract: The
More informationin 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
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Brown representability in A 1 -homotopy theory Niko Naumann and Markus Spitzweck Preprint Nr. 18/2009 Brown representability in A 1 -homotopy theory Niko Naumann and Markus
More informationUniversity of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor
Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.
More informationCATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)
CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.
More informationA CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 15 11 A CLASSIFICATION THEOREM FOR NORMAL EXTENSIONS MATHIEU DUCKERTS-ANTOINE AND TOMAS EVERAERT Abstract: For a particular
More informationarxiv: v1 [math.ct] 27 Oct 2017
arxiv:1710.10238v1 [math.ct] 27 Oct 2017 Notes on clans and tribes. Joyal October 30, 2017 bstract The purpose o these notes is to give a categorical presentation/analysis o homotopy type theory. The notes
More informationCategory Theory. Course by Dr. Arthur Hughes, Typset by Cathal Ormond
Category Theory Course by Dr. Arthur Hughes, 2010 Typset by Cathal Ormond Contents 1 Types, Composition and Identities 3 1.1 Programs..................................... 3 1.2 Functional Laws.................................
More informationGALOIS CATEGORIES MELISSA LYNN
GALOIS CATEGORIES MELISSA LYNN Abstract. In abstract algebra, we considered finite Galois extensions of fields with their Galois groups. Here, we noticed a correspondence between the intermediate fields
More informationTHE COALGEBRAIC STRUCTURE OF CELL COMPLEXES
Theory and pplications o Categories, Vol. 26, No. 11, 2012, pp. 304 330. THE COLGEBRIC STRUCTURE OF CELL COMPLEXES THOMS THORNE bstract. The relative cell complexes with respect to a generating set o coibrations
More informationC2.7: CATEGORY THEORY
C2.7: CATEGORY THEORY PAVEL SAFRONOV WITH MINOR UPDATES 2019 BY FRANCES KIRWAN Contents Introduction 2 Literature 3 1. Basic definitions 3 1.1. Categories 3 1.2. Set-theoretic issues 4 1.3. Functors 5
More informationMath Homotopy Theory Hurewicz theorem
Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S
More informationHomotopy Theory of Topological Spaces and Simplicial Sets
Homotopy Theory of Topological Spaces and Simplicial Sets Jacobien Carstens May 1, 2007 Bachelorthesis Supervision: prof. Jesper Grodal KdV Institute for mathematics Faculty of Natural Sciences, Mathematics
More informationGOURSAT COMPLETIONS DIANA RODELO AND IDRISS TCHOFFO NGUEFEU
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 18 06 GOUSAT COMPLETIONS DIANA ODELO AND IDISS TCHOFFO NGUEFEU Abstract: We characterize categories with weak inite
More informationCategory Theory. Categories. Definition.
Category Theory Category theory is a general mathematical theory of structures, systems of structures and relationships between systems of structures. It provides a unifying and economic mathematical modeling
More informationIn the index (pages ), reduce all page numbers by 2.
Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More information3 3 Lemma and Protomodularity
Ž. Journal o Algebra 236, 778795 2001 doi:10.1006jabr.2000.8526, available online at http:www.idealibrary.com on 3 3 Lemma and Protomodularity Dominique Bourn Uniersite du Littoral, 220 a. de l Uniersite,
More informationarxiv: v3 [math.at] 28 Feb 2014
arxiv:1101.1025v3 [math.at] 28 Feb 2014 CROSS EFFECTS AND CALCULUS IN AN UNBASED SETTING (WITH AN APPENDIX BY ROSONA ELDRED) KRISTINE BAUER, BRENDA JOHNSON, AND RANDY MCCARTHY Abstract. We studyunctors
More informationLOCALIZATIONS, COLOCALIZATIONS AND NON ADDITIVE -OBJECTS
LOCALIZATIONS, COLOCALIZATIONS AND NON ADDITIVE -OBJECTS GEORGE CIPRIAN MODOI Abstract. Given two arbitrary categories, a pair of adjoint functors between them induces three pairs of full subcategories,
More informationRepresentable presheaves
Representable presheaves March 15, 2017 A presheaf on a category C is a contravariant functor F on C. In particular, for any object X Ob(C) we have the presheaf (of sets) represented by X, that is Hom
More informationThe Ordinary RO(C 2 )-graded Cohomology of a Point
The Ordinary RO(C 2 )-graded Cohomology of a Point Tiago uerreiro May 27, 2015 Abstract This paper consists of an extended abstract of the Master Thesis of the author. Here, we outline the most important
More informationSome glances at topos theory. Francis Borceux
Some glances at topos theory Francis Borceux Como, 2018 2 Francis Borceux francis.borceux@uclouvain.be Contents 1 Localic toposes 7 1.1 Sheaves on a topological space.................... 7 1.2 Sheaves
More informationTopos Theory. Lectures 21 and 22: Classifying toposes. Olivia Caramello. Topos Theory. Olivia Caramello. The notion of classifying topos
Lectures 21 and 22: toposes of 2 / 30 Toposes as mathematical universes of Recall that every Grothendieck topos E is an elementary topos. Thus, given the fact that arbitrary colimits exist in E, we can
More informationAlgebraic Geometry
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationVariations on a Casselman-Osborne theme
Variations on a Casselman-Osborne theme Dragan Miličić Introduction This paper is inspired by two classical results in homological algebra o modules over an enveloping algebra lemmas o Casselman-Osborne
More informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationCategorical models of homotopy type theory
Categorical models of homotopy type theory Michael Shulman 12 April 2012 Outline 1 Homotopy type theory in model categories 2 The universal Kan fibration 3 Models in (, 1)-toposes Homotopy type theory
More informationCombinatorial Models for M (Lecture 10)
Combinatorial Models for M (Lecture 10) September 24, 2014 Let f : X Y be a map of finite nonsingular simplicial sets. In the previous lecture, we showed that the induced map f : X Y is a fibration if
More informationALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.
ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)
More informationLogarithm of a Function, a Well-Posed Inverse Problem
American Journal o Computational Mathematics, 4, 4, -5 Published Online February 4 (http://www.scirp.org/journal/ajcm http://dx.doi.org/.436/ajcm.4.4 Logarithm o a Function, a Well-Posed Inverse Problem
More informationDerivatives of the identity functor and operads
University of Oregon Manifolds, K-theory, and Related Topics Dubrovnik, Croatia 23 June 2014 Motivation We are interested in finding examples of categories in which the Goodwillie derivatives of the identity
More informationNotes on Beilinson s How to glue perverse sheaves
Notes on Beilinson s How to glue perverse sheaves Ryan Reich June 4, 2009 In this paper I provide something o a skeleton key to A.A. Beilinson s How to glue perverse sheaves [1], which I ound hard to understand
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationHOMOLOGICAL DIMENSIONS AND REGULAR RINGS
HOMOLOGICAL DIMENSIONS AND REGULAR RINGS ALINA IACOB AND SRIKANTH B. IYENGAR Abstract. A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the
More informationON PROPERTY-LIKE STRUCTURES
Theory and Applications o Categories, Vol. 3, No. 9, 1997, pp. 213 250. ON PROPERTY-LIKE STRUCTURES G. M. KELLY AND STEPHEN LACK Transmitted by R. J. Wood ABSTRACT. A category may bear many monoidal structures,
More informationLECTURE 6: FIBER BUNDLES
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann
More informationAn introduction to locally finitely presentable categories
An introduction to locally finitely presentable categories MARU SARAZOLA A document born out of my attempt to understand the notion of locally finitely presentable category, and my annoyance at constantly
More information3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection
3. Categories and Functors We recall the definition of a category: Definition 3.1. A category C is the data of two collections. The first collection is called the objects of C and is denoted Obj(C). Given
More information