Category Theory. Lectures by Peter Johnstone Notes by David Mehrle Cambridge University Mathematical Tripos Part III Michaelmas 2015

Transcription

1 Category Theory Lectures by Peter Johnstone Notes by David Mehrle Cambridge University Mathematical Tripos Part III Michaelmas 2015 Contents 1 Introduction The Yoneda Lemma djunctions Limits and Colimits Monads Filtered Colimits belian and dditive Categories Last updated June 5,

2 Contents by Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture

3 1 Introduction 1 Introduction Category theory has been around or about hal a century now, invented in the 1940 s by Eilenberg and MacLane. Eilenberg was an algebraic topologist and MacLane was an algebraist. They realized that they were doing the same calculations in dierent areas o mathematics, which led them to develop category theory. Category theory is really about building bridges between dierent areas o mathematics. 1.1 Deinitions and examples This is just about setting up the terminology. There will be no theorems in this chapter. Deinition 1.1. category C consists o (i) a collection ob C o objects, B, C,... (ii) a collection mor C o morphisms, g, h... (iii) two operations, called domp q and codp q, rom morphisms to objects. We write ÝÑ B or : Ñ B or P mor C and domp q and codp q B; (iv) an operation ÞÑ 1 rom objects to morphisms, such that 1 ÝÑ ; (v) an operation : p, gq ÞÑ g rom pairs o morphisms (so long as we have dom cod g) to morphisms, such that domp gq dompgq and codp gq codp q. These data must satisy: (vi) or all : Ñ B, 1 1 B ; (vii) composition is associative. I g and gh are deined, then pghq p gqh. Remark 1.2. (a) We don t require that ob C and mor C are sets. (b) I they are sets, then we call C a small category. (c) We can get away without talking about objects, since ÞÑ 1 is a bijection rom ob C to the collection o morphisms satisying g g and h h whenever teh composites are deined. Essentially, we can represent objects by their identity arrows. Example 1.3. (a) The category Set whose objects are sets and whose arrows are unctions. Technically, we should speciy the codomain or the unctions because really the deinition o a unction doesn t speciy a codomain. So morphisms are pairs p, Bq, where B is the codomain o the unction. Lecture October 2015

4 1 Introduction 1.1 Deinitions and examples (b) Gp is the category o groups and group homomorphisms; (c) Ring is the category o rings and ring homomorphisms; (d) R-Mod is the category o R-modules and R-module homomorphisms; (e) Top is the category o topological spaces and continuous maps; () M is the category o smooth maniolds and smooth maps; (g) The homotopy category o topological spaces Htpy has the same objects as Top, but the morphisms X Ñ Y are homotopy classes o continuous maps; (h) or any category C, we can turn the arrows around to make the opposite category C op. Example (h) leads to the duality principle, which is a kind o two or the price o one deal in category theory. Theorem 1.4 (The Duality Principle). I φ is a valid statement about categories, so is the statement φ obtained by reversing all the morphisms. Example (g) above gives rise to the ollowing deinition. Deinition 1.5. In general, an equivalence relation on the collection o all morphisms o a category is called a congruence i (i) g ùñ dom dom g and cod cod g; (ii) g ùñ h gh and k kg whenever the composites are deined. There s a category C{ with the same objects as C but -equivalence classes as morphisms. Example 1.6. Continued rom Example 1.3. (i) category C with one object must have dom cod or all P mor C. So all composites are deined, and (i mor C is a set), mor C is just a monoid (which is a semigroup with identity). (j) In particular, a group can be considered as a small category with one object, in which every morphism is an isomorphism. (k) groupoid is a category in which all morphisms are isomorphisms. For a topological space X, the undamental groupoid πpxq is the basepointless undamental group; the objects are points o X and the morphisms x Ñ y are homotopy classes paths rom x to y. (Homotopy classes are required so that each path has an inverse). (l) a category whose only morphisms are identites is called discrete. category in which, or any two objects, B there is at most one morphism Ñ B is called preorder, i.e. it s a collection o objects with a relexive and transitive relation. In particular, a partial order is a preorder in which the only isomorphisms are identities. Lecture October 2015

5 1 Introduction 1.1 Deinitions and examples (m) The category Rel has the same objects as Set, but the morphisms are relations instead o unctions. Precisely, a morphism Ñ B is a triple p, R, Bq where R Ď ˆ B. The composite pb, S, Cqp, R, Bq is p, R S, Cq where R S tpa, cq Db P B s.t. pa, bq P R and pb, cq P Su. Note that Set is a subcategory o Rel and Rel Rel op. Let s continue with the examples. Example 1.7. Continued rom Example 1.3. (n) Let K be a ield. The category Mat K has natural numbers as objects. morphism n Ñ p is a p ˆ n matrix with entries in K. Composition is just matrix multiplication. Note that, once again, Mat K Mat op k, via transposition o matrices. (o) n example rom logic. Suppose you have some ormal theory T. The category Det T o derivations relative to T has ormulae in the language o T as objects, and morphisms φ Ñ ψ are derivations φ ψ and composition is just concatenation. The identity 1 φ is the one-line derivation φ. Deinition 1.8. Let C and D be categories. unctor F : C Ñ D consists o (i) an operation ÞÑ Fpq rom ob C to ob D; (ii) an operation ÞÑ Fp q rom mor C to mor D, satisying (i) dom Fp q Fpdom q, cod Fp q Fpcod q or all ; (ii) Fp1 q 1 Fpq or all ; (iii) and Fp gq Fp qfpgq whenever g is deined. Let s see some examples again. Example 1.9. (a) The orgetul unctor Gp Ñ Sets which sends a group to its underlying set, and any group homomorphism to itsel as a unction. Similarly, there s one Ring Ñ Set, and Ring Ñ b, and Top Ñ Set (b) There are lots o constructions in algebra and topology that turn out to be unctors. For example, the ree group construction. Let F denote the ree group on a set. It comes equipped with an inclusion map Lecture October 2015

6 1 Introduction 1.1 Deinitions and examples η : Ñ F, and any : Ñ G, where G is a group, extends uniquely to a homomorphism F Ñ G. F G η F is a unctor rom Set to Gp, and given g : Ñ B, we deine Fg to be the unique homomorphism extending the composite ÝÑ g B η B ÝÑ FB. (c) The abelianization o an arbitrary group G is the quotient G{G 1 o G by it s derived subgroup G 1 xxyx 1 y 1 x, y P Gy. This gives the largest quotient o G which is abelian. I φ : G Ñ H is a homomorphism, then it maps the derived subgroup o G to the derived subgroup o H, so the abelianization is unctorial Gp Ñ b. (d) The powerset unctor. For any set, let P denote the set o all subsets o. P is a unctor Set Ñ Set; given : Ñ B, we deine P p 1 q t pxq x P 1 u or 1 Ď. But we also make P into a unctor P : Set Ñ Set op (or Set op Ñ Set) by setting P pb 1 q 1 pb 1 q or B 1 Ď B. This last example is what we call a contravariant unctor. Deinition contravariant unctor F : C Ñ D is a unctor F : C Ñ D op (equivalently, C op Ñ D). The term covariant unctor is used sometimes to make it clear that a unctor is not contravariant. Example Continued rom Example 1.9 (e) The dual space o a vector space over K deines a contravariant unctor k-mod Ñ k-mod. I α : V Ñ W is a linear map, then α : W Ñ V is the operation o composing linear maps W Ñ K with α. () Let Cat denote the category o small categories and unctors between them. Then C ÞÑ C op is covariant unctor Cat Ñ Cat. (g) I M and N are monoids, regarded as one-object categories, what is a unctor between them? It s just a monoid homomorphism rom M to N: it preserves the identity element and composition. In particular, i M, N are groups, then the unctor is a group homomorphism. Hence, we may think o Gp is a subcategory o Cat. (h) Similarly, i P and Q are partially ordered sets, regarded as categories, a unctor P Ñ Q is just an order-preserving map. (i) Let G be a group, regarded as a category. unctor F : G Ñ Sets picks out a set as the image o the one object in G, and each morphism o G is an isomorphism so gets mapped to a bijection o this set. So this is a group action G FpGq. I we replace Sets by k-vect or k a ield, we get linear representations o G. œ Lecture October 2015

7 1 Introduction 1.1 Deinitions and examples (j) In algebraic topology, there are many unctors. For example, the undamental group π 1 px, xq deines a unctor rom Top (the category o pointed topological spaces, i.e., those with a distinguished basepoint) to Gp. Similarly, homology groups are unctors H n : Top Ñ b (or more commonly, Htpy Ñ b). There s another layer too. There are morphisms between unctors, called natural transormations. Deinition Let C and D be categories and F, G : C Ñ D. natural transormation α : F Ñ G is an operation ÞÑ α rom ob C to mor D, such that dompα q Fpq, codpα q Gpq or all, and the ollowing diagram commutes. F F FB α G G α B GB gain, we should mention some examples o natural transormations. Example (a) There s a natural transormation α : 1 k-mod Ñ, where is the dual space unctor. This is the statement that a vector space is canonically isomorphic to it s double dual. α V : V Ñ V sends r P V to the evaluate at r map V Ñ k. I we restrict to inite-dimensional spaces, then α becomes a natural isomorphism, i.e. an isomorphism in the category rk-gmod, k-gmods, where rc, Ds denotes the category o all unctors C Ñ D with natural transormations as arrows. Remark Note that i α is a natural transormation, and each α is an isomorphism, then the inverses β o the α also orm a natural transormation, because β B G β B G α β β B α B F β F β. Example Continued rom Example 1.13 (b) Let F : Sets Ñ Gp be the ree group unctor, and let U : Gp Ñ Set be the orgetul unctor. The inclusion o generators η : Ñ UF is the -component o a natural transormation 1 Set Ñ UF. (c) For any set, the mapping a ÞÑ tau is a unction t u : Ñ Ppq. We see that t u is a natural transormation 1 Set Ñ P, since or any : Ñ B, we have P ptauq t paqu. (d) Suppose given two groups G, H and two homomorphisms, 1 : G Ñ H. natural transormation Ñ 1 is an element h P H such that h pgq 1 pgqh or all g P G, or equivalently, h pgqh 1 1 pgq. So such a transormation exists i and only i and 1 are conjugate. Lecture October 2015

8 1 Introduction 1.1 Deinitions and examples (e) For any space X with a base point x, there s a natural homomorphism h px,xq : π 1 px, xq Ñ H 1 pxq called the Hurewicz homomorphism. This is the px, xq-component o the natural transormation h rom π 1 to the composite U Top ÝÑ Top H 1 ÝÑ b ÝÑ I Gp, where U is the orgetul unctor and I is the inclusion. It s not oten useul to say that unctors are injective or surjective on objects. Generally, a unctor might output some object which is isomorphic to a bunch o others, but might not actually be surjective it could be surjective up to isomorphism. This is is similar to the idea that equality is not useul when comparing groups, but rather isomorphism. Deinition Let F : C Ñ D be a unctor. We say F is (1) aithul i, given, g P mor C, the three equations domp q dompgq, codp q codpgq, and F Fg imply g; (2) ull i, given g : F Ñ FB in D, there exists : Ñ B in C with F g. We say a subcategory C 1 o C is ull i the inclusion unctor C 1 Ñ C is ull. Example (a) b is a ull subcategory o Gp; (b) The category Lat o lattices (that is, posets with top element 1, bottom element 0, binary joint _, binary meet ^) is a non-ull subcategory o Posets. Likewise, equality o categories is a very rigid idea. Isomorphism o categories, as well, is a little bit too rigid. We might have several objects in a category C which are isomorphic in C and all mapped to the same object in D in this case, we want to consider these categories somehow the same. I we require isomorphism o categories, we cannot insist on even the number o objects being the same. See Example 1.20 or a concrete realization o this. Deinition Let C and D be categories. n equivalence o categories between C and D is a pair o unctors F : C Ñ D and G : D Ñ C together with natural isomorphisms α : 1 C Ñ GF, β : FG Ñ 1 D. The notation or this is C» D. Deinition We say that a property o categories is a categorical property i whenever C has property P and C» D, then D has P as well. Example (a) Given an object B o a category C, we write C{B or the category whose objects are morphisms ÝÑ B with codomain B, and whose morphisms Lecture October 2015

9 1 Introduction 1.1 Deinitions and examples g : p ÝÑ Bq ÝÑ p 1 1 ÝÑ Bq are commutative triangles g 1 1 B The category Sets{B is equivalent to the category Sets B o B-indexed amilies o sets. In one direction, we send p ÝÑ Bq to p 1 pbq b P Bq, and in the other direction we send pc b b P Bq to ď C b ˆ tbu π 2 ÝÑ B. bpb Composing these two unctors doesn t get us back to where we started, but it does give us something clearly isomorphic. (b) Let 1{Set be the category o pointed sets p, aq, and let Part be the subcategory o Rel whose morphisms are partial unctions, i.e. relations R such that pa, bq P R and pa, b 1 q P R implies b b 1. Then 1{Set» Part: in one direction we send p, aq to ztau and : p, aq Ñ pb, bq to tpx, yq x P, y P B, pxq y, y bu In the other direction we send to p Y tu, q and a partial unction ã B (apparently that s the notation or partial unctions) to the unction deined by $ & paq a P dom paq B a P z dom % B a (c) The category dmod k o inite dimensional vector spaces over k is equivalent to dmod op k by the dual unctors dmod k dmod op k, and the natural isomorphism 1 dmodk Ñ. This is an equivalence but not an isomorphism o categories. (d) The category dmod k is also equivalent to Mat k : in one direction send an object o n o Mat k to k n, and a morphism to the linear map with matrix relative to the standard basis. In the other direction, send a vector space V to dim V and choose a basis or each V to send a linear transormation θ : V Ñ W to the matrix representing θ with respect to the chosen bases. The composite Mat k Ñ dmod k Ñ Mat k is the identity; the other composite is isomorphic to the identity via the isomorphisms sending the chosen bases to the standard basis o k dim V. Lecture October 2015

10 1 Introduction 1.1 Deinitions and examples There s another notion slightly weaker than surjectivity o a unctor. Some call it surjective up to isomorphism. Deinition We say a unctor F : C Ñ D is essentially surjective i or every object D o D, there exists an object C o C such that D FpCq. The next lemma somehow uses a more powerul version o the axiom o choice and is beyond usual set theory. Lemma unctor F : C Ñ D is part o an equivalence between C and D i and only i F is ull, aithul, and essentially surjective. Proo pùñq. Suppose given G : D Ñ C, α : 1 C Ñ GF and β : FG Ñ 1 D as in the deinition o equivalence o categories. Then B FGB or all B, so F is clearly essentially surjective. Let s prove aithulness. Now suppose given, g : Ñ B P C such that F Fg. Then GF GFg. Using the naturality o α, the ollowing diagram commutes: Now GF GF GFg GFB α α B (1) B α 1 B pgf qα α 1 B pgfgqα g, the last line by the naturality o α with g along the bottom arrow o (1) instead o. Thereore, is aithul. For ullness, suppose given g : F Ñ FB in D. Deine α 1 B pggq α : Ñ GF Ñ GFB Ñ B. Observe that Gg α B α 1. Similarly, the ollowing square commutes: GF α GF GFB α B B by the naturality o α. Thereore, GF α B α 1 GF α B α 1 Gg. as well. Hence, pplying the argument or aithulness o F to the unctor G shows that G is aithul. Thereore, GF Gg implies that F g. Hence, the unctor F is ull. pðùq. We have to deine the unctor G : D Ñ C. Lecture October 2015

11 1 Introduction 1.1 Deinitions and examples For each object D o D, there is some C P ob C such that FC D, because F is essentially surjective. Deine GD C or some choice o C, and also choose an isomorphism β D : FGD Ñ D or each D. This deines G on objects. To deine G on morphisms, suppose we are given g : X Ñ Y in D. Since F is ull and aithul, there is a unique : GX Ñ GY such that F g. We could deine G in this way, but we won t because we want β to be a natural isomorphism. Instead, we deine Gg : GX Ñ GY to be the unique morphism in C whose image under F is FGX β X X g β 1 Y Y FGY. This deinition guarantees that β is a natural isomorphism. To check that G is a unctor, we can apply aithulness o F to assert that Gpgq Gphq Gpghq whenever gh is deined, since these two morphisms o C have the same image under F. So G is a unctor, and β is a natural isomorphism FG Ñ 1 D by construction. To deine α : Ñ GF, take it to be the unique map in C such that Fα β 1 F. This is an isomorphism since the unique morphism mapped to β F is a two-sided inverse or it. We just need to check that α is a natural transormation. Given a (not necessarily commuting) square α GF GF B α B GFB (2) apply F to it to get the square F F FB Fα β 1 F FGF FGF FGFB Fα B β 1 FB which commutes by the naturality o β. In particular, this gives us the equality Fpα B q Fα B F β 1 FB F FGF β 1 F FGF Fα FpGF α q Faithulness o F then implies that α B GF α, so in particular the square (2) commutes. Hence, α is a natural transormation. Deinition (a) category C is skeletal i or any isomorphism in C, domp q codp q; Lecture October 2015

12 1 Introduction 1.1 Deinitions and examples (b) By a skeleton o a category C, we mean a ull subcategory C 1 containing exactly one object rom each isomorphism class o C. Note that by Lemma 1.22, i C 1 is a skeleton o C then the inclusion unctor C 1 Ñ C is part o an equivalence. Remark More or less any statement you make about skeletons o small categories is equivalent to the (set-theoretic) axiom o choice, including each o the ollowing statements: (a) Every small category has a skeleton; (b) Every small category is equivalent to any o it s skeletons; (c) ny two skeletons o a given small category are isomorphic. How do we discuss morphisms within a category being surjective and injective? The correct category-theoretic generalization o these notions are epimorphism and monomorphism. Deinition morphism : Ñ B is a category C is (a) a monomorphism, or monic, i, given any g, h : C Ñ with g h, we have g h; (b) an epimorphism, or epic, i given any k, l : B Ñ D with k l, we have k l. We write : B to indicate that is monic, and : B to indicate that is epic. category C is called balanced i every P mor C which is both monic and epic is an isomorphism. Example (a) In Set, is monic i and only i injective, and is epic i and only i surjective. Thereore, Set is a balanced category. (b) In Gp, monomorphisms are injective and epimorphisms are surjective (nontrivial to show that every epimorphism is surjective). So Gp is balanced. (c) In Rings, every monomorphism is injective, but not all epimorphisms are surjective. For example, the inclusion map Z Q is both monic and epic, but clearly not an isomorphism. (d) In Top, monic and epic are equivalently injective and surjective. But Top isn t balanced. (e) In a poset, every morphism is both monic and epic, so the only balanced posets are discrete ones. Lecture October 2015

13 2 The Yoneda Lemma 2 The Yoneda Lemma Remark 2.1. It may seem odd that we re devoting a whole ew lectures to just one lemma, but Yoneda s Lemma is really much more. It s an entire way o thinking about category theory! In act, it s both much more and much less than a lemma. Remark 2.2. The Yoneda Lemma should probably not be attributed to Yoneda, because he never wrote it down! It was discovered by many category theorists in the early work on the subject, but who irst wrote it down we don t know. It s called the Yoneda lemma because Saunders MacLane attributed it to Yoneda in his book, but the paper he cited doesn t contain the lemma! In the next edition, MacLane instead attributed it to private correspondence, which means Yoneda told it to him once while they were waiting or a train. Deinition 2.3. We say a category C is locally small i, or any two objects, B, the morphisms Ñ B in C orm (are parameterized by) a set Cp, Bq, which is sometimes written Hom C p, Bq. Deinition 2.4. Let C be a locally small category. Given P ob C, we have a unctor Cp, q: C Ñ Set sending B to Cp, Bq, and a morphism g : B Ñ C to the pullback unction g : Cp, Bq Ñ Cp, Cq given by g p q g. This is unctorial by associativity o composition in C. This is the covariant representable unctor. Similarly we can make ÞÑ Cp, Bq into a unctor Cp, Bq: C op Ñ Set, which is called the contravariant representable unctor. Lemma 2.5 (Yoneda Lemma). Let C be a locally small category, let be an object o C, and let F : C Ñ Set be a unctor. Then (i) there is a bijection between natural transormations Cp, q Ñ F and elements o F; and (ii) the bijection in (i) is natural in both F and. Remark 2.6. Note that we haven t assumed that C is a small category! So the content o Lemma 2.5 transcends set theory. Proo o Lemma 2.5. (i) We have to construct a bijection between the set F and natural transormations Cp, q Ñ F. To that end, given any natural transormation α : Cp, q Ñ F, we deine Φpαq α p1 q, which is an element o F. Conversely, given x P F, we deine Ψpxq: Cp, q Ñ F by Ψpxq B : Cp, Bq FB F pxq Lecture October 2015

14 2 The Yoneda Lemma To veriy that Ψpxq is genuinely a natural transormation, let g : B Ñ C be a morphism in C, and then consider the diagram: Cp, Bq FB Ψpxq B Cp,gq Fg Cp, Cq FC ΨpXq C (3) Let P Cp, Bq. Then going around the diagram (3) clockwise, Ψpxq C pcp, gqp qq Ψpxq C pg q Fpg qpxq and going around (3) counterclockwise, Fpgq pψpxq B p qq Fpgq Fp qpxq Fpg qpxq. This veriies that Ψpxq is a natural transormation, or each x P F. We should check that Ψ and Φ are inverses. So: ΦΨpxq Ψpxqp1 q Fp1 qpxq 1 F pxq x. ΨpΦpαqq B p q Fp qpφpαqq Fp qpα p1 qq α B pcp, qp1 qq α B p 1 q α B p q or all B and : Ñ B, so ΨpΦpαqq α. Thereore, Φ and Ψ are inverse. (ii) I C is small (so that rc, Sets is locally small) then we have two unctors: C ˆ rc, Sets p, Fq Set F C ˆ rc, Sets p, Fq Set rc, SetspCp, q, Fq where rc, SetspCp, q, Fq is a conusing notation or the set o natural transormations between Cp, q and F. The assertion o (ii) is that Φ is a natural isomorphism among these unctors. For naturality in, suppose given : 1 Ñ. We want to show that the ollowing square commutes: rc, SetspCp 1, q, Fq F 1 Φ 1 Θ F rc, SetspCp 1, q, Fq F Φ (4) Lecture October 2015

15 2 The Yoneda Lemma where Θpαq α Cp, q. Suppose we are given α : Cp 1, q Ñ F. We will chase the image o α around the diagram (4) in two dierent ways, and show they are equal. Going counterclockwise, and going clockwise, F pφpαqq F pα 1p1 1qq α p q ΦpΘpαqq Φpα Cp, qq α pcp, q p1 1qq α p q This veriies that Φ is natural in. To show that Φ is natural in F, suppose given a natural transormation η : F Ñ G. We want to show that the ollowing diagram commutes: rc, SetspCp, q, Fq F Φ F η rc, SetspCp, q, Gq η G Φ G (5) Once again, let α : Cp, q Ñ F and chase the diagram (5) around counterclockwise η pφ F pαqq η pα p1 qq and clockwise Φ G pη αq pη αq p1 q η pα p1 qq to see that it commutes. Hence, Φ is natural in F as well. Corollary 2.7. The unctor Y : C op Ñ rc, Sets given by ÞÑ Cp, q is ull and aithul. Hence, every locally small category is equivalent to a subcategory o a unctor category rc, Sets. Proo. Note that or : B Ñ, Yp q Cp, q. By Lemma 2.5(i), natural transormations Cp, q Ñ CpB, q correspond bijectively to elements o CpB, q. But it s not clear that this bijection comes rom the map Y; i it does, then Y is ull and aithul. So we want to show that Y is the inverse o Φ : rc, SetspCp, q, CpB, qq Ñ CpB, q, where Φ is the natural transormation as in Lemma 2.5 To that end, let α : Cp, q Ñ CpB, q be a natural transormation. Then YpΦpαqq Ypα p1 qq Cpα p1 q, q is a natural transormation. Now or any g : Ñ C, YpΦpαqq C pgq Cpα p1 q, Cqpgq g α p1 q. (6) Lecture October 2015

16 2 The Yoneda Lemma Because α is a natural transormation, the ollowing square commutes. Cp, q g Cp, Cq α α C CpB, q g CpB, Cq Chasing this diagram starting with 1 in the top-let, we see that g α p1 q α C pg 1 q α C pgq Thereore, substituting into (6), we have YpΦpαqq C pgq g α p1 q α C pgq. This holds or any C and any g : Ñ C, so it ollows that YpΦpαqq α, so Y is a let-inverse to Φ. Since Φ is a bijection, this means that Y is a right-inverse to Φ as well, and thereore Y is also a bijection between CpB, q and natural transormations Cp, q Ñ CpB, q. Deinition 2.8. We call this unctor Y : C op Ñ rc, Sets the Yoneda Embedding. This is not unlike the Cayley representation theorem in group theory, which says that every inite group is a subgroup o a symmetric group. In act, the Cayley representation theorem is just a special case o the Yoneda embedding! Deinition 2.9. We say a unctor C Ñ Set is representable i it s isomorphic to Cp, q or some. By a representation o F : C Ñ Set, we mean a pair p, xq with x P F such that Ψpxq is a natural isomorphism Cp, q Ñ F. We also call x a universal element o F. Corollary 2.10 (Representations are unique up to unique isomorphism). I p, xq and pb, yq are both representations o F, then there s a unique isomorphism : Ñ B such that F pxq y. Proo. The composite CpB, q ÝÝÝÑ Ψpyq F ÝÝÝÝÑ Ψpxq 1 Cp, q is an isomorphism, so it s o the orm Cp, q or a unique isomorphism : Ñ B by Corollary 2.7. So the ollowing diagram commutes. Cp, q CpB, q Cp, q Ψpyq F Ψpxq In particular, plug in B to this diagram CpB, Bq Cp,Bq Cp, Bq Ψpyq B FB Ψpxq B Lecture October 2015

17 2 The Yoneda Lemma and chase 1 B P CpB, Bq around the diagram in two ways: pψpxq B Cp, Bqqp1 B q Ψpxq B p1 B q Ψpxq B p q F pxq pψpxq B Cp, Bqqp1 B q Ψpyq B p1 B q Fp1 B qpyq 1 FB pyq y Thereore, F pxq y. Oten we will abuse terminology and talk about the representation o a unctor, but this is okay by Corollary 2.10, because any two representations are uniquely isomorphic. Representable unctors appear everywhere, as the next example shows. Example (a) The orgetul unctor Gp Ñ Set is representable by pz, 1q; (b) The orgetul unctor Ring Ñ Set is representable by pzrxs, xq; (c) The orgetul unctor Top Ñ Set is representable by pt u, q. (d) The contravariant power-set unctor P : Set op Ñ Set is representable by pt0, 1u, t1, uq since there s a natural bijection between subsets 1 Ď and unctions χ : Ñ t0, 1u. (e) The dual-vector-space unctor k-mod op Ñ k-mod, when composed with the orgetul unctor k-mod Ñ Set, is representable by pk, 1 k q. () For a group G, the unique (up to representation) representable unctor G Ñ Set is the let-regular representation o G, that is, G acts on the set G with let-multiplication. This is the Cayley representation theorem o group theory. Note that the endomorphisms o this object o rg, Sets are just the right multiplications h ÞÑ hk or some ixed k P G. So they orm a group isomorphic to G op. Deinition Given two objects, B o a locally small category C, we can orm the unctor Cp, q ˆ Cp, Bq: C op Ñ Set. representation o this unctor is a (categorical) product o and B. What does this look like? It consists o an object ˆ B and two maps π 1 : ˆ B Ñ, and π 2 : ˆ B Ñ B such that, given any pair p : C Ñ, g : C Ñ Bq, there s a unique x, gy: C Ñ ˆ B such that π 1 x, gy and π 2 x, gy g. Deinition Given a parallel pair o maps g B, the assignment EpCq th : C Ñ h ghu. deines a subunctor E o Cp, q. representation o E, i it exists, is called an equalizer o and g: it consists o an object E and a morphism e : E Ñ such that e ge such that every h : C Ñ with h gh actors uniquely through e. e E B g C h Lecture October 2015

18 2 The Yoneda Lemma Remark I e : E Ñ is an equalizer o the two maps, g : Ñ B, then it s necessarily monic, since any morphism h : C Ñ actors through e in at most one way. In particular, given a, b : C Ñ E such that ea eb, we have that ea actors through e in at most one way. One such way that it does actor is as written: e a. nother such way is e b, but the way it actors must be unique so it must be that a b. a E B C b e g Deinition We call a monomorphism : B (i) regular i it occurs as an equalizer; (ii) split i there is g : B Ñ with g 1. Lemma (i) split monomorphism is regular monic. (ii) morphism which is epic and regular monic is an isomorphism. Proo. (i) Let be split monic with let-inverse g. Claim that is the equalizer o g and 1 B. I g 1 then g 1 1 B. nd i h : C Ñ B satisies gh 1 B h, then h actors through via gh. I h k is another such actorization, then gh h k implies that k gh because is monic, so the actorization is unique. Hence is an equalizer. gh B B C h (ii) Suppose e : E Ñ is epic and an equalizer o the maps, g : Ñ B. Then e eg, which means that g because e is epic. We know that 1 is another map such that 1 g1, so the map 1 actors through e as ek 1 or some map k : Ñ E. Thereore, e has a right-inverse and so is split epic. The dual o statement (i) shows that a split epimorphism is a regular epimorphism, so e is both monic and regular epic. The dual o the argument in the previous paragraph then produces a let-inverse l : Ñ E such that le 1 E, and moreover g 1 B l l1 lek 1 E k k so the inverses are equal. Hence, e is an isomorphism. Deinition Let C be a category and let G be a collection o objects in C. Lecture October 2015

19 2 The Yoneda Lemma (a) We say G is a separating amily i, whenever we have that, g : Ñ B in C such that h gh or all h : G Ñ with G P G, then g. (b) We say G is a detecting amily i given : Ñ B such that every morphism h : G Ñ B with G P G actors uniquely through, then is an isomorphism. (c) I G tgu is a singleton, we call G a separator or detector or C, depending on which case we re in. Remark Here is an equivalent deinition o separating and detecting amilies or a locally small category C. amily G is separating i and only i the collection o unctors tcpg, q G P Gu are jointly aithul and G is detecting i and only i tcpg, q G P Gu jointly relect isomorphisms. Lemma (i) I C has equalizers, then every detecting amily or C is separating. (ii) I C is balanced (mono + epi ùñ iso), then every separating amily is detecting. Proo. (i) Suppose that G is a detecting amily and we have maps, g : Ñ B such that i h gh or all h : G Ñ or all G P G, then g. Let e : E Ñ be an equalizer o and g; then every h : G Ñ with G P G actors uniquely through e, so e is an isomorphism and thereore g. e G E B (ii) Suppose G is a separating amily and : Ñ B satisying the hypotheses o Deinition 2.17(b). I k, l : B Ñ C satisy k l, then kh lh or all h : G Ñ B with G P G, so k l. Hence, is epic. Similarly, i p, q : D Ñ satisy p q, then or any n : G Ñ D we have pn qn, so both pn and qn are actorizations o pn through, and hence equal. Because G is a separating amily, this means in turn that p q. Hence, is monic. By the assumption that C is balanced, is thereore an isomorphism. Example (a) ob C is always both separating and detecting. (b) I C is locally small, then tcp, q P ob Cu is both separating and detecting or rc, Sets. h g Lecture October 2015

20 2 The Yoneda Lemma (c) Z is a separator and a detector or Gp. The unctor that it represents is the orgetul unctor Gp Ñ Set, and that unctor is aithul and respects isomorphisms. (d) In all sensible algebraic categories, the ree object on one generator is both separator and detector. (e) t u is a separator, but not a detector, or Top. In act, Top has no detecting amily since or any cardinal κ, we can ind a set X and topologies τ 1 Ĺ τ 2 on X that agree on any subset o X with cardinality less than κ. lso Top op has a detector, namely X tx, y, zu with topology tx, H, tx, yuu. Deinition We say an object P is projective in C i, whenever we re given g e P B with e epic, then there is g : P Ñ with eg. Dually, P is injective i it s projective on C op. I the condition holds not or all epimorphisms e but or some class E o epis, we say that P is E-projective. (For example, i E is the regular epimorphisms, then P is regular-projective). This deinition generalizes the algebraic notion o projective objects. Lemma Representable unctors are projective as elements o the unctor category. More precisely, or any locally small C, the unctors Cp, q are all E-projective in rc, Sets, where E is the class o pointwise-surjective natural transormations. We can t prove this or E the set o all epimorphisms o rc, Sets yet, because we don t know what epimorphisms are in this category. But it will turn out that they re exactly the pointwise-surjective natural transormations, so what we prove below suices. Proo o Lemma Use the Yoneda lemma. Given the diagram Cp, q F β G α let y P G correspond to α. Then there is x P F with β pxq y, and the corresponding Ψpxq: Cp, q Ñ F satisies β Ψpxq α. Making the analogy with algebra, this says that the unctors Cp, q are kind o like the ree objects o rc, Sets (insoar as ree objects are projective). Lecture October 2015

21 3 djunctions 3 djunctions This theory was irst developed by D.M. Kan in the paper djoint Functors which appeared in TMS in The real diiculty in ormalizing the idea that had been around or a ew years was inding the right level o abstraction: maximally useul but suiciently general. The deinition we give is the one that Kan gave in that paper. Deinition 3.1. Let C and D be categories and F : C Ñ D, G : D Ñ C be two unctors. n adjunction between F and G is a natural bijection between morphism F Ñ B in D and morphisms Ñ GB in C, or all P ob C and B P ob D. I C and D are locally small, this is a natural isomorphism between the unctors DpFp q, q and Cp, Gp qq. These are both unctors C op ˆ D Ñ Set. Notice that the deinition o adjunction has a deinite direction: we say that F is let-adjoint to G or that G is right-adjoint to F, and write F % G. Example 3.2. Examples o djunctions (a) The ree group unctor F : Set Ñ Gp is let-adjoint to the orgetul unctor U : Gp Ñ Set. For any set and any group G, each unction Ñ UG extends to a unique homomorphism F Ñ G, and this correspondence is natural. (b) The orgetul unctor U : Top Ñ Set has a let adjoint D, where D is with the discrete topology. ny unction Ñ UX becomes continuous as a map D Ñ X. U also has a right adjoint I, which is I with the indiscrete topology t, Hu. ny map into an indiscrete space is continuous, so any map UX Ñ is continuous as a map X Ñ I. So we have adjunctions D % U % I. (c) Consider the unctor ob: Cat Ñ Set. This has a let adjoint given by the discrete category unctor D : Set Ñ Cat, where D is the discrete category with the objects same as elements o ; any mapping Ñ ob C deines a unique unctor D Ñ C. In particular, D % ob. It also has a right adjoint I, which is the indiscrete unctor. I has objects those elements o with one morphism a Ñ b or each pa, bq P ˆ. gain, a unctor C Ñ I is determined by its eect on objects. ob % I D has a let adjoint the connected components unctor π 0, where π 0 C pob Cq{ is the smallest equivalence relation such that U V whenever there exists : U Ñ V in C. ny unctor C Ñ D is constant on each -equivalence class, but any unction π 0 C Ñ can occur. (d) Let Idem be the category whose objects are pairs p, eq, or some set and idempotent e : Ñ, e 2 e. The morphisms : p, eq Ñ p 1, e 1 q in this category are those unctions satisying e 1 e. Lecture October 2015

22 3 djunctions Let G : Idem Ñ Set be the unctor sending p, eq to ta P epaq au and a morphism to its restriction Gp,eq. Let F : Set Ñ Idem be the unctor sending to p, id q. Now F % G since any morphism F Ñ pb, eq takes values in GpB, eq. nd urthermore, G % F since any morphism p, eq ÝÑ FB is determined by its eect on Gp, eq, since paq pepaqq or all a P Gp, eq. (e) For any C, there s a unique unctor C Ñ 1 (where 1 is the category with one object and one morphism). let adjoint or this speciies an object I o C such that there s a unique morphism I Ñ or any, i.e. an initial object o C. Dually, a right adjoint to C Ñ 1 speciies a terminal object. () Let and B be sets and : Ñ B. Then we have order-preserving maps P : P Ñ PB and P : PB Ñ P and P % P since P p 1 q Ď B 1 i and only P 1, pxq P B 1 i and only i Ď P pb 1 q. (g) Suppose given two sets, B and a relation R Ď ˆ B. We deine orderreversing maps L : PB Ñ P by LpB 1 q ta P B 1, pa, bq P Ru R : P Ñ PB by RpB 1 q tb P P 1, pa, bq P Ru For any subsets 1, B 1 we have 1 Ď LpB 1 q ô 1 ˆ B 1 Ď R ô B 1 Ď Rp 1 q. We say a pair pf, Gq o contravariant C Ø D are adjoint on the right i F : C Ñ D op % G : D op Ñ C. (h) The contravariant power-set unctor P is sel-adjoint on the right, since unctions Ñ PB correspond to subsets R Ď ˆ B, and hence to unctions B Ñ P. Given a unctor G : D Ñ C, we want to know i it possibly has a let adjoint. To that end, we deine a category p Ó Gq called the comma category (because it was originally written p, Gq with a comma instead o an arrow) Deinition 3.3. The comma/arrow category p Ó Gq has objects all pairs pb, q with B P ob D and : Ñ GB. The morphisms pb, q Ñ pb 1, 1 q are morphisms g : B Ñ B 1 in D which make the diagram below commute. 1 Gg GB GB 1 Theorem 3.4. Speciying a let-adjoint or G is equivalent to speciying an initial object o p Ó Gq or each P ob C. Lecture October 2015

23 3 djunctions Proo. pùñq Suppose F % G. For any, F 1 F ÝÝÑ F corresponds to a morphism η : Ñ GF. I claim that pf, η q is initial in p Ó Gq. Given an object pb, q in p Ó Gq, the assertion that η GF commutes is equivalent to saying that Gg GB F F 1 F g B commutes, where corresponds to under the adjunction. So the unique morphism pf, η q Ñ pb, q is. Last time we were in the middle o proving Theorem 3.4. We proved one direction last time, so let s start by inishing the proo. Proo o Theorem 3.4, continued. pðùq. Suppose given an initial object pb, η q in p Ó Gq, or each. Deine F B or all objects o C. To deine F on morphisms, let : Ñ 1 in C. We deine F : F Ñ F 1 to be the unique morphism F Ñ F 1 making the ollowing commute. η η 1 GF GF 1 GF 1 The uniqueness o this morphism comes rom the act that pf, η q is initial. This uniqueness ensures that F is unctorial: i 1 : 1 Ñ 2, then pf qpf 1 q and Fp 1 q both it into the same naturality square or η, and so by the uniqueness o morphisms that it into this square, they re equal. Furthermore, by construction, this makes η : 1 Ñ GF a natural transormation. Given a morphism y : Ñ GB, we know that there is a unique map x : F Ñ B such that the ollowing commutes: η y GF GB Gx In particular, this gives a bijection between morphisms x : F Ñ B and morphisms y : Ñ GB sending x : F Ñ B to Φpxq pghqη. To show this Lecture October 2015

24 3 djunctions is natural in B, suppose given g : B Ñ B 1. We want to show the ollowing commutes. Φ DpF, Bq Cp, GBq g DpF, Bq Φ Gg Cp, GBq But this ollows because or any h : F Ñ B, we have Φpghq Gpghqη pggqpghqη pggqφphq. Naturality in ollows rom the act that η is natural. More precisely, given : 1 Ñ, we want to show that the ollowing commutes DpF, Bq F DpF 1, Bq Φ Φ Cp, GBq Cp 1, GBq But this is true because or any h : F Ñ B, we have Φph F q Gph F q η Gphq GFp q η Gphq η Φphq. This way o thinking about adjoints turns out to be quite useul. Corollary 3.5. I F and F 1 are both let adjoints or G, then F is naturally isomorphic to F 1. Proo. Let η be the map Ñ GF that corresponds to id F : F Ñ F, and likewise let η 1 be the map Ñ GF1 that corresponds to id F 1 : F 1 Ñ F 1. Then pf, η q and pf 1, η 1 q are both initial in p Ó Gq, which means that they must be isomorphic as objects in this category. hence, there is a unique isomorphism α : pf, η q Ñ pf 1, η 1 q that is natural in : given : Ñ 1, then there two ways around the naturality square F F α α 1 F 1 F 1 F 1 F 1 1 are both morphisms pf, η q Ñ pf 1 1, Gpα 1q η 1 q in p Ó Gq, and this morphism is unique because pf, η q is initial. Lemma 3.6. Given unctors F C D E with F % G and H % K, we have HF % GK. G H K Lecture October 2015

25 3 djunctions Proo. Given P ob C, C P ob E, we have bijections EpHF, Cq ÐÑ DpF, KCq and DpF, KCq ÐÑ Cp, GKCq. Compose these bijections to get the result. Corollary 3.7. Suppose given a commutative square o categories and unctors C E H F K D F G in which all the unctors have let adjoints. Then the diagram o let-adjoints commutes up to natural isomorphism. Proo. The two ways round it are both let-adjoint to GF KH by Lemma 3.6, so they re isomorphic by Corollary 3.5. Example 3.8. unctor with a right adjoint preserves initial objects, i they exist. I F : C Ñ D % G : D Ñ C, then the diagram below commutes. D G C 1 But a let adjoint or C Ñ 1 picks out an initial object o C by Example 3.2(e). So F maps it to an initial object o D. Theorem 3.9. Suppose given F : C Ñ D and G : D Ñ C. Speciying an adjunction F % G is equivalent to speciying two natural transormations η : 1 C Ñ GF and ε : FG Ñ 1 D such that F Fη FGF 1 ε F and F F G η G GFG 1 G Gε (7) G both commute. η and ε are called the unit and counit o the adjunction, and the two diagrams in Equation 7 are called the triangular identities. Proo. pùñq. Given F % G, with a natural bijection Θ,B : DpF, Bq Ñ Cp, GBq, deine η Θ,F p1 F q and ε B Θ 1 GB,B p1 GBq We want to show that both η and ε are natural in and B, respectively. Note that the deinition o ε is dual to η, so it suices to check that η is natural and the naturality o ε will ollow dually. To check the naturality o η, suppose given : Ñ 1. We want to show that η GF η 1 GF 1 GF 1 Lecture October 2015

26 3 djunctions commutes. We can use the naturality o Θ to show this. In particular, consider the ollowing diagram DpF, Fq F DpF, F 1 q Θ,F Θ,F 1 Cp, GFq GF Cp, GF 1 q which commutes by the naturality o Θ. Chase 1 F around the diagram starting in the upper let to see that Θ,F 1pF q GFp q. 1 F Θ,F p1 F q η (8) F 1 F Θ,F 1pF q GFp q η Now what is Θ,F 1pF q? To answer this question, consider the diagram DpF, F 1 q Θ,F 1 Cp, GF 1 q F DpF 1, F 1 q Θ 1,F 1 Cp 1, GF 1 q which again commutes by naturality o Θ. Now chase 1 F 1 around this diagram, starting in the lower let. 1 F 1 F Θ,F 1pF q η 1 (9) 1 F 1 η 1 Θ 1,F 1p1 F 1q Thereore, combining (8) and (9), we get that η is natural, because η 1 Θ,F 1pF q GF η. Finally, it remains to check the triangular identities. We can only check one o them; the other ollows dually. Let g : B Ñ B 1 in D and let : F Ñ B. By commutativity the ollowing diagram (which ollows again rom naturality o Θ) DpF, Bq Θ,B Cp, GBq g DpF, B 1 q Θ,B 1 Gg Cp, GB 1 q we can conclude that Θpg q Gg Θp q. In particular, or B FG, 1 F, g ε : FG Ñ, we conclude one o the triangular identities. 1 GB ΘpΘ 1 p1 GB qq Θpε B q Θpε B 1 FGB q Gε B Θ 1FGB Gε B η GB Lecture October 2015

27 3 djunctions The other one ollows dually. pðùq. Conversely, suppose given η and ε satisying the triangular identities. We need to establish a natural bijection between maps F Ñ B and Ñ GB or P C, B P D. Given h : F Ñ B, deine φphq pghqη : Ñ GF Ñ GB, and given k : Ñ GB, deine ψpkq ε B pfkq: F Ñ FGB Ñ B. These are natural in both and B (check this!) and they re inverse to each other by the triangular identities: and similarly or φ ψpkq. ψφphq ε B pfφphqq ε B pfghqfη hε F BFη h, Is every equivalence o categories C F G D, with maps α : 1 C ÝÑ FG and β : FG ÝÑ 1 D an adjunction? The answer to this question is yes-and-no, assuming certain conditions. Lemma 3.10 (Every Equivalence is an adjoint equivalence). Suppose given F, G, α, β as above. Then there exist isomorphisms α 1 : 1 C ÝÑ GF, β 1 : FG ÝÑ 1 D satisying the triangular identities. In particular, F % G and G % F. Proo. Deine α 1 α and let β 1 be the composite β 1 : FG ÝÝÑ FGβ 1 pfαgq 1 FGFG ÝÝÝÝÝÑ FG ÝÑ β 1 D. Note that FGFG FGβ FG FG β FG β β 1 D commutes by the naturality o β, and β is monic, so FGβ β FG. Now β 1 Fα pfgβfq 1 β 1 FGF FFα F ÝÑ FGF ÝÝÝÝÝÝÝÝÝÝÑ FGFGF Fα GF ÝÝÝÑ FGF β F ÝÑ F We can rewrite this by β 1 F F ÝÝÑ FGF ÝÝÝÑ FGFα pfαgfq 1 pfgfαq 1 FGFGF ÝÝÝÝÝÝÝÝÝÝÝÝÝÑ FGF β F ÝÑ F, and so everything cancels! Thereore, β 1 F Fα 1 F. To see the other way, Gβ 1 α 1 G is the composite Gβ 1 α 1 G G α G ÝÑ GFG ÝÝÝÝÝÝÑ pgfgβq 1 pgfαgq 1 GFGFG ÝÝÝÝÝÝÑ GFG ÝÝÑ Gβ G G ÝÝÝÝÑ pgβq 1 GFG α GFG pαgfgq 1 ÝÝÝÑ GFGFG ÝÝÝÝÝÝÑ GFG ÝÝÑ Gβ G 1 G Lecture October 2015

28 3 djunctions Lemma Let G : D Ñ C be a unctor having a let adjoint F, with counit ε : FG Ñ 1 D. Then (i) G is aithul ðñ ε is (pointwise) epic. (We say (pointwise) epic because it will turn out that arrows in a unctor category are epic i they are pointwise epic. But we don t know that yet). (ii) G is ull and aithul ðñ ε is an isomorphism. Proo. (i) Let : B Ñ B 1 be a morphism in D. The composite ε B : FGB Ñ B Ñ B 1 corresponds under the adjunction to G : GB Ñ GB 1. So ε B is an epimorphism or all B ðñ or all B, B 1, composition with ε B is an injection DpB, B 1 q Ñ DpFGB, B 1 q ðñ or all B, B 1, application o G is an injection DpB, B 1 q Ñ CpGB, GB 1 q ðñ G is aithul. (ii) Similarly, G is ull and aithul ðñ or all B, composition with ε B is a bijection DpB, B 1 q Ñ DpFGB, B 1 q. This clearly holds i ε B is an isomorphism or all B. Conversely, i G is ull and aithul, then 1 FGB ε B or some : B Ñ FGB. By (i), we know that ε B is epic and this shows that ε B is split monic. Hence, ε B is an isomorphism. Deinition (a) n adjunction pf % Gq is called a relection i G is ull and aithul. (b) We say that a ull subcategory C 1 Ď C is relective i the inclusion C 1 Ñ C has a let adjoint. This comes with a caveat, that this terminology isn t ully standard. Some people don t require that a relective subcategory is ull, but I think it makes more sense to talk about relective subcategories when they correspond to the relections. Example (a) b is relective in Gp: the let adjoint to the inclusion sends G to it s abelianization G{G 1. (b) n abelian group is torsion i all o its elements have inite order. In any abelian group, the elements o inite order orm a subgroup called the torsion subgroup T, and any homomorphism B Ñ where B is torsion takes values in T. So ÞÑ T deines a corelection rom b to the ull subcategory b T o torsion groups. Similarly, ÞÑ { T deines relection to the subcategory o b that consists o torsion-ree groups. Lecture October 2015

29 4 Limits and Colimits (c) There many examples o this in topology, and the most important o these is the Stone-Čech compactiication. Let KHaus Ď Top be the ull subcategory o compact Hausdor spaces. The inclusion KHaus Ñ Top has a let adjoint β, called the Stone-Čech compactiication. Interestingly, Stone and Čech gave dierent constructions o the compactiication that now has their name, and essentially the only way to show that these two constructions are equal is to show that they are both let-adjoint to the orgetul unctor. 4 Limits and Colimits To talk about limits, we need to ormally deine what a diagram is. We ve been drawing diagrams but we don t quite yet know what they are. Deinition 4.1. Let J be a category (usually small, and oten inite). diagram o shape J in C is a unctor D : J Ñ C. Example 4.2. I J I J commute., then a diagram o shape J is a commutative square in C., then a diagram o shape J is a square in C that need not Deinition 4.3. Given D : J Ñ C, a cone over D consists o an object C P ob C (the apex o the cone) together with morphisms λ j : C Ñ Dpjq (the legs o the cone) or each j P ob J such that C λ j Dpiq commutes or all α : j Ñ j 1 in J. λ i Dpαq Dpjq I we write C or the constant diagram o shape J sending all j P ob J to C and all α : j Ñ j 1 to 1 C, then a cone over D with apex C is a natural transormation C Ñ D. Then is a unctor C Ñ rj, Cs. Deinition 4.4. (a) The category o cones over D is the arrow/comma category p Ó Dq, deined dually to Theorem 3.4. (b) The category o cones under D is the arrow/comma category pd Ó q, as deined in Theorem 3.4. Lecture October 2015

30 4 Limits and Colimits Deinition 4.5. By a limit or D : J Ñ C, we mean a terminal object o p Ó Dq. colimit or D is an initial object o pd Ó q. We say that C has limits (resp. colimits) o shape J i : C Ñ rj, Cs has a right (resp. let) adjoint. Example 4.6. (a) Suppose J H. Then there s a unique D : H Ñ C, and p Ó Dq C. So a limit (resp. colimit) or D is a terminal (resp. initial) object o C. In Set, 1 t u is terminal and H is initial. Similarly in Top. In Gp, the trivial group t1u is both initial and terminal, and in Ring Z is initial. (b) Let J be the discrete category with two objects. diagram o shape J is a pair o objects p 1, 2 q, a limit or this is a product 1 ˆ 2 π 1 π 2 and a colimit or J is a coproduct i 1 i 2 1 ` 2 In Set, Gp, Ring, Top,..., the product are cartesian products (with suitable structure). In Set and Top, corproducts are disjoint unions 1 \ 2. In Gp, coproducts are ree products G 1 G 2. In b, inite coproducts coincide with inite products. Last time we were talking about limits and colimits, and giving some examples. We saw the product o two objects, but we can also deine the product o many objects. Deinition 4.7. (i) More generally, let J be any (small) discrete category. diagram o shape J is a J-indexed amily o objects p j j P Jq. limit or it is a product ś jpj ś j equipped with projections π i : jpj j Ñ i. Dually, a coproduct ř jpj j with ν i : i Ñ ř jpj j. (ii) Let J be the category Ñ. diagram o shape J is a parallel pair Ñ g B; a cone over it looks like a diagram x C λ 1 λ 2 g B Lecture October 2015