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 of category theory. Contents 1 Categories, functors and natural transformations 1 1.1 Objects and morphisms................................. 1 1.2 Universal properties................................... 2 1.3 Functors.......................................... 4 2 Representable functors 6 2.1 The Yoneda Lemma................................... 6 2.2 Representations of functors............................... 7 3 Limits and colimits 8 3.1 Categories of (co)cones.................................. 8 3.2 Having limits....................................... 11 4 Adjunctions 13 4.1 Preservation of limits................................... 15 4.2 Kan extensions...................................... 16 References 17 Please send comments and corrections to cnewstead@cmu.edu.
1 Categories, functors and natural transformations 1.1 Objects and morphisms Definition 1.1. A category C is a collection C 0 of objects and a collection C 1 of morphisms such that: ˆ Each morphism f has a domain A and a codomain B, denoted f : A B; ˆ If f : A B and g : B C then there is a morphism g f : A C, and moreover the operation is associative; ˆ Each object A has an identity morphism id A : A A, which satisfies f id A = f and id A g = g for all f : A B and g : C A. Given objects A and B, the collection of morphisms A B is denoted Hom C (A, B). Definition 1.2. A morphism f : A B in a category C is... ˆ... a monomorphism if f x = f y x = y for all x, y : C A, in which case we say f is monic; ˆ... an epimorphism if u f = v f u = v for all u, v : B D, in which case we say f is epic; ˆ... an isomorphism if it has an inverse g : B A, which satisfies g f = id A and f g = id B. If f is an isomorphism then its inverse is unique, since if g and h are two inverses then We denote the inverse of f by f 1. Definition 1.3. g = g id B = g (f h) = (g f) h = id A h = h ˆ An object A in a category C is initial if for each object B there is a unique morphism A B. Typically we make a choice of initial object (which is sometimes canonical) and denote it 0. ˆ An object A in a category C is terminal if for each object C there is a unique morphism C A. Typically we make a choice of terminal object (which is sometimes canonical) and denote it 1. Some examples of categories are: ˆ The category Set of sets and functions between them. In this category, the monomorphisms are the injective functions, the epimorphisms are the surjective functions, and the isomorphisms are the bijections. The empty set is the (only) initial object and singletons are terminal objects. 1
ˆ The category Top of topological spaces and continuous maps between them. In this category, the monomorphisms and epimorphisms are the injective and surjective maps and the isomorphisms are the homeomorphisms. Like in Set, the empty space is the initial object and the singleton space is the terminal object. ˆ The category Gp of groups and group homomorphisms between them. In this category, the monomorphisms and epimorphisms are the injective and surjective homomorphisms, and the isomorphisms are the group isomorphisms. The trivial group is both initial and terminal in Gp. ˆ Any poset P = (P, P ) can be viewed as a category whose objects are the elements of P. Given any two objects a, b P, there is a unique morphism a b in P if and only if a P b. Initial objects are P -minimal elements, and terminal objects are P -maximal elements. Only identity morphisms are isomorphisms, since a P b and b P a if and only if a = b. All morphisms are monic and epic because morphisms are determined by their domain and codomain. ˆ Any directed graph G = (V, E) generates a category G, whose objects are the vertices of G and whose morphisms are identities and formal composites of edge relations. ˆ Given a category C there is an opposite category C op. C op has the same objects as C, and a morphism B A in C op is precisely a morphism A B in C. I will typically denote by f : A B the morphism in C op given by the morphism f : A B in C. 1.2 Universal properties Many concepts in category theory are defined by universal properties, which characterise an object up to isomorphism by the way it interacts with other objects. Some examples are given by the following definitions. Definition 1.4. Given objects A and B in a category C, a product of A and B is an object P together with projection morphisms π A : P A and π B : P B such that, whenever f : X A and g : X B, there is a unique morphism u : X P such that π A u = f and π B u = g. This is illustrated in the following diagram: f X u g A P B π A π B We often make a choice of product for a given pair A, B and denote it by A B; and in this case we write u = f, g. Some examples of products include: 2
ˆ In Set, Top and Gp, the product of A and B is the usual cartesian product A B (endowed with the appropriate structure in the case of Top and Gp). ˆ In a poset P, p is a product of a and b if and only if it is a (the) greatest lower bound of a and b, i.e. a b = a b. Definition 1.5. Given objects A and B in a category C, a coproduct of A and B is an object C together with inclusion morphisms ι A : A C and ι B : B C such that, whenever f : A Y and g : B Y, there is a unique morphism u : C Y such that u ι A = f and u ι B = g. This is illustrated in the following diagram: ι A ι B A C B f Y u g We often make a choice of product for a given pair A, B and denote it by A + B; and in this case we write u = [f, g]. Some examples of coproducts include: ˆ In Set and Top, the coproduct of A and B is the disjoint union A B, with ι A : A A B and ι B : B A B the canonical inclusion maps. ˆ In Top, the category of pointed topological spaces (topological spaces with a specified basepoint and basepoint-preserving continuous functions), the coproduct of A and B is the wedge union A B, obtained by taking the disjoint union of A and B and identifying their basepoints. ˆ In a poset P, c is a product of a and b if and only if it is a (the) least upper bound of a and b, i.e. a + b = a b. Definition 1.6. Let A, B, C be objects and f : A C and g : B C be morphisms in a category C. The pullback of (f, g) is an object P together with morphisms p A : P A and p B : P B such that f p A = g p B and, whenever q A : Q A and q B : Q B with f q A = g q B, there is a unique morphism u : Q P such that q A = p A u and q B = p B u. This is illustrated in the following diagram: Q u q A P p A A q B p B f B g C 3
In Set we can define P = {(a, b) A B : f(a) = g(b)} Definition 1.7. Let A, B, C be objects and f : C A and g : C B be morphisms in a category C. The pushout of (f, g) is an object P together with morphisms p A : A P and p B : B P such that p A f = p B g and, whenever q A : A Q and q B : B Q with q A f = q B g, there is a unique morphism u : P Q such that q A = u p A and q B = u p B. This is illustrated in the following diagram: C g B f p B q B A p A P u q A Q In Set we can define P = A B/ where is the smallest equivalence relation satisfying f(c) g(c) for c C. 1.3 Functors Definition 1.8. Let C and D be categories. A (covariant) functor F : C D is a map of objects and of arrows such that F (id A ) = id F (A) for all objects A C 0 and F (g f) = F (g) F (f) for all composable pairs of morphisms f, g in C. A contravariant functor from C to D is a covariant functor C op D. Some examples of functors are: ˆ Given any category C there is an identity functor id C morphisms to themselves. : C C sending all objects and ˆ The forgetful functor U : Top Set, sending each space to its underlying set and each continuous map to its underlying function. ˆ The discrete topology functor D : Set Top, which sends each set to the discrete space on that set, and each function to the (continuous) map between these spaces. ˆ Given two posets P and Q, a functor F : P Q is precisely an order-preserving map. The collection of small 1 categories and functors between them forms a category, denoted by Cat. 1 A category is small if the collection of all its objects and morphisms is a set. 4
Definition 1.9. Let C and D be categories and F, G : C D be functors. A natural transformation α : F G is a collection of D-morphisms α A : F (A) G(A), for A C 0, such that for each C-morphism f : A B the following diagram commutes: F (f) F (A) F (B) α A α B G(A) G(B) G(f) We will sometimes indicate that there is a natural transformation F G by declaring that a morphism F (A) G(A) is natural in A. Some examples of natural transformations include: ˆ Given any functor F : C D there is an identity natural transformation id F : F F, defined by (id F ) A = id F (A). ˆ Consider the functor P : Set Set defined by sending a set A to its powerset P(A) and a function f : A B to the direct image map f : P(A) P(B). There is a natural transformation {} : id Set P given by {} A (x) = {x} for all x A. To see that this is natural, let f : A B and x A; then so {} B f = f {} A. ({} B f)(x) = {f(x)} = f ({x}) = (f {} A )(x) ˆ Let Vec k be the category of vector spaces over a field k and linear maps and let : (Vec k ) op Vec k be the dual functor, which sends a space V to its dual V = L(V, k) and a linear map f : V W to its dual f : W V, defined by f (α) = α f. There is a natural transformation θ : id Veck ( ) defined by θ V (v)(α) = α(v) for α V. To see that this is natural, let f : V W, v V and α W ; then (θ W f)(v)(α) = α(f(v)) = (α f)(v) = f (α)(v) = (θ V (v) f )(α) = (f θ V )(v)(α) so θ W f = f θ V. Given categories C, D there is a category [C, D], whose objects are functors C D and whose morphisms are natural transformations. We say a natural transformation α : F G is a natural isomorphism if it is an isomorphism in [C, D]. Definition 1.10. A functor F : C D is an equivalence of categories if there is a functor G : D C such that G F = id C and F G = id D. We say C and D are equivalent and write C D. 5
2 Representable functors 2.1 The Yoneda Lemma Definition 2.1. A category C is locally small if, for any objects A and B, the collection Hom C (A, B) is a set. For example, Set is locally small because given any two sets A, B there is only a set of functions A B. However Cat is not locally small because given any two categories C, D there may be a proper class of functors C D: for instance, for each set A there is a functor ( ) A : Set Set. Definition 2.2. If C is locally small then there is a functor y : C [C op, Set] defined by y(a)(x) = Hom C (X, A), y(a f B) X (X g A) = f g for X C 0 and g : X A. The functor y is called the Yoneda embedding. Theorem 2.3 (Yoneda lemma). Let C be a locally small category, F : C op Set be a functor and A C 0. Then there is a bijection which is natural in A and in F. Hom [C op,set](y(a), F ) = F (A) Proof. Define maps Φ A,F : Hom [C op,set](y(a), F ) F (A) : Ψ A,F by Φ A,F (α) = α A (id A ), Ψ A,F (x) C (C g A) = F (g)(x) We ll drop subscripts which are fixed in our portion of working. Φ and Ψ are an inverse pair. For x F (A) we have Φ(Ψ(x)) = Ψ(x) A (id A ) = F (id A )(x) = id F (A) (x) = x so Φ Ψ = id F (A). For α : y(a) F, C C 0 and g : C A we have Ψ(Φ(α)) C (g) = F (g)(φ(α)) = F (g)(α A (id A )) = α C (g id A ) = α C (g) so Ψ(Φ(α)) = α and Ψ Φ = id Hom[C op,set] (y(a),f ). Φ is natural in A. F : C D: We need the following diagram to commute for all f : A B and all Hom [C op,set](y(a), F ) ( ) f Hom [C op,set](y(b), F ) Φ A Φ B F (A) F (B) F (f) 6
Let θ : y(b) F. Then and so the diagram commutes. (F (f) Φ B )(θ) = F (f)(θ B (id B )) = (F (f) θ B )(id B ) = θ A (f) Φ A (θ f) = (θ f) A (id A ) = θ A (f id A ) = θ A (f) Φ is natural in F. We need the following diagram to commute for all A C 0 and all natural transformations η : F G: Hom [C op,set](y(a), F ) η ( ) Hom [C op,set](y(b), F ) Φ F Φ G F (A) η A F (B) Let θ : y(a) F. Then so the diagram commutes. (η A Φ F )(θ) = η A (θ A (id A )) = (η θ) A (id A ) = Φ G (η θ) Corollary 2.4. Let C be locally small and F : C op Set be a functor. For all C-objects A and B, A = B if and only if y(a) = y(b) Proof. Apply the Yoneda Lemma (2.3) to the functor y(b). We see the Yoneda lemma all over the place. For instance, it tells us that for a set A, elements of A correspond naturally with maps { } A. But this is obvious: a A corresponds with the function ǎ : { } A given by ǎ( ) = a. It also tells us (via the above corollary) that, in a poset category P, a = b if and only if c P a c P b for all c P. 2.2 Representations of functors Definition 2.5. Let C be locally small and F : C op Set be a functor. We say F is representable if there is a natural isomorphism θ : y(a) = F for some A C 0. By the Yoneda lemma (2.3) θ corresponds with an element x F (A). The pair (A, x) is called a representation of F. For example, the forgetful functor U : Gp Set is representable by ((Z, +), 1), since every group homomorphism G Z is determined by the image of 1. Proposition 2.6. Suppose F : C op Set is representable and that (A, x) and (B, y) are two representations of F. Then there is a unique isomorphism f : A B in C such that F f(y) = x. 7
3 Limits and colimits 3.1 Categories of (co)cones Definition 3.1. Let C and J be categories. A diagram of shape J in C is a functor d : J C. We call J the index category of the diagram. We often identify a diagram with its image. For example, a diagram of shape in a category C is a pair of morphisms P = A f B g C For future reference, we ll call this diagram d : P C. Definition 3.2. Let d : J C be a diagram. A cone over d is an object C together with morphisms ε i : C d(i) for i J 0 such that d(α) ε i = ε j for each α : i j in J. That is, the following diagram commutes for all α: ε i d(i) C d(α) ε j d(j) For example, a cone over the diagram P given above consists of an object Q together with morphisms q A : Q A, q B : Q B and q C : Q C such that the following diagram commutes: Q q A q B q C A f B g C We may omit q C from the picture since q C = f q A = g q B. 8
Definition 3.3. Let d : J C be a diagram of shape J in a category C, and let (C, ε i ) and (D, δ i ) be cones over d. A morphism of cones (C, ε i ) (D, δ i ) is a C-morphism m : C D such that δ i m = ε i for all i J 0. Thus the following diagram commutes for all α : i j C ε i d(i) δ i m D d(α) δ j ε j d(j) Given a diagram d : J C there is a category Cone(d) whose objects are cones over d and whose morphisms are morphisms of cones. Definition 3.4. A terminal cone (L, λ i ) over d is called a limit of d, often denoted lim(d). For example, a limit of the diagram d : P C above is a cone A p A P p B B such that, given any other cone A q A Q q B B, there is a unique morphism u : Q P making the following diagram commute: Q u q A P p A A q B p B f B g C That is, a limit of a diagram of shape is precisely a pullback. Other examples of limits include: ˆ A limit of a diagram of shape 0 (the empty category) is a terminal object. ˆ A limit of a diagram of shape is simply an object. ˆ A limit of a diagram of shape is a product (of a pair). ˆ More generally, a limit of a diagram of shape D λ, where D λ is the discrete category with λ-many objects (and only identity morphisms), is a product. 9
ˆ A limit of a diagram of shape is an equalizer. In Set the equalizer of f, g : A B is the inclusion e : E A of the subset E = {a A : f(a) = g(a)} Dually we have the notions of a cocone under d, a morphism of cocones, the category of cocones (denoted Cocone(d)) and a colimit (denoted colim(d)), summarised by the following diagram: d(i) ε i d(α) λ i λ j colim(d) u C d(j) ε j That is, a colimit is an initial object in the category of cocones. Examples of limits include: ˆ A colimit of a diagram of shape 0 (the empty category) is an initial object. ˆ A colimit of a diagram of shape is simply an object. ˆ A colimit of a diagram of shape is a coproduct (of a pair). ˆ More generally, a colimit of a diagram of shape D λ is a coproduct. ˆ A colimit of a diagram of shape is an coequalizer. In Set the coequalizer of f, g : A B is the quotient map q : B Q where Q is the quotient of B by the smallest equivalence relation identifying f(a) and g(a) for each a A. ˆ A colimit of a diagram of shape is a pushout. ˆ A colimit in Top of a sequence of inclusions of spaces X 0 X 1 X 2 is precisely their union X = n X n, where U X is open in X n if U X n is open for all n. Since they re defined by universal properties, limits and colimits are unique up to isomorphism. Before moving on I will give another example of limits and colimits that give something quite familiar: 10
Example 3.5. Let G be a group considered as a category, i.e. G 0 = { } and G 1 is the set of elements of the group considered as morphisms with group multiplication interpreted as composition. (So all morphisms in G are isomorphisms.) A functor ρ : G Set is precisely a group action on the set ρ( ) =: X defined by g x = ρ(g)(x) for g G 1 and x X. Indeed, given g, h G 1 and x X we have (gh) x = ρ(gh)(x) = ρ(g)(ρ(h)(x)) = g (h x) A cone over ρ is a set C equipped with a morphism i : C X such that for all g G 1 we have ρ(g) i = i. A limit, if it exists, will be a morphism λ : 1 X (i.e. an element λ X such that ρ(g)(λ) = λ for all g, i.e. a fixed point for the action. A cocone under ρ is a set Q equipped with a morphism j : X Q such that for all g G 1 we have j ρ(g) = j. The colimit of ρ always exists and is the set X/G of orbits; indeed if q : X X/G is the quotient map then we have [x] = [g x] for any g, which is precisely the statement that q ρ(g) = q. 3.2 Having limits Definition 3.6. A category C... ˆ... has (co)limits of shape J if every diagram d : J C has a (co)limit in C; ˆ... has all finite (co)limits if it has (co)limits of shape J for every finite category J ; ˆ... has all small (co)limits if it has (co)limits of shape J for every small category J ; ˆ... has all (co)limits if it has (co)limits of shape J for every category J. It is useful to find sufficient conditions for having (co)limits of various shapes and sizes. Theorem 3.7. Let C be a category. If C has equalizers and all (small, finite) products then C has all (small, finite) limits. Proof. Let d : J C be a diagram (with J small, finite). Define P = d(j), Q = d(j) j J 0 α:i j and let π j : P d(j) and ρ α : Q d(j) denote the product projections. Define morphisms f, g : P Q by f = π j α : i j, g = d(α) π i α : i j that is, f is the unique morphism P Q with ρ α f = π j for each α : i j, and so on. 11
Finally, construct the equalizer L e P f g and define λ i = π i e : L d(i) for each i J 0. Claim. (L, λ i ) = lim(d). First we need to check it is a cone. Well if α : i j in J then d(α) λ i = d(α) π i e = ρ α g e = ρ α f e = π j e = λ j Now suppose (C, ε i ) is another cone over d, so that d(α) ε i = ε j for all α : i j in J. Since P is a product, there is a unique morphism x : C P such that π j x = ε j for each j J 0. But then f x = π j x α : i j = ε j α : i j = d(α) ε i α : i j = d(α) π i x α : i j = d(α) π i α : i j x = g x Since L equalizes f and g there is a unique morphism u : L C such that x u = e. But then for each i J 0. So (L, λ i ) = lim(d). Q ε i u = π i x u = π i e = λ i Theorem 3.8. Suppose C has a terminal object and pullbacks. Then C has all finite limits. Proof. It suffices to show that C has products of pairs of objects and equalizers of pairs of objects. Products of pairs. Let A, B C 0. Let A π A P π B B make the following diagram a pullback: P π A A π B! A B! B 1 where! A,! B are the unique morphisms to the terminal object 1. Any morphisms A f X g B satisfy! A f =! B g =! X, so for any such morphisms there is a unique arrow u : X P such that π A u = f and π B u = g. Thus P is the product of A and B. Equalizers. Let f, g : A B. Form the pullback e E e A id A, f A A B id A, g 12
Since the diagram commutes, we have e = π A id A, f e = π A id A, g e = e so f e = g e = g e. Since the diagram is a pullback, e : E A is universal with this property, so E e A is the equalizer of f and g. By the previous theorem, C has all finite limits. Corollary 3.9. Let F : C D be a functor. ˆ If C has and F preserves equalizers and all (small, finite) products then F preserves all (small, finite) limits. ˆ If C has and F preserves terminal objects and pullbacks then F preserves all finite limits. 4 Adjunctions An adjunction is a relation between functors that appears all over the place in category theory and has some important properties. Definition 4.1. Let C and D be locally small categories and F : C D and G : D C be functors. We say F is left-adjoint to G, and G is right-adjoint to F, if for all C-objects A and D-objects B there is a bijection Φ A,B : Hom D (F (A), B) Hom C (A, G(B)) which is natural in A and B. If F is left-adjoint to G we write F G; the relation is called an adjunction. This definition is only valid for locally small categories. If we re willing to abuse what we mean by bijection then this definition also extends to general categories, but we can do better. As such, I will define an adjunction again, and then prove that the two definitions are equivalent. Definition 4.2. Let C and D be categories and F : C D and G : D C be functors. We say F is left-adjoint to G, and G is right-adjoint to F, if there exist natural transformations η : id C G F (called the unit of the adjunction) and ε : F G id D (called the counit of the adjunction) satisfying the following triangular identities: F F η F G F G ηg G F G εf Gε id F F id G G 13
where εf is the natural transformation with components (εf ) A = ε F (A) and so on. Theorem 4.3. Definitions 4.1 and 4.2 are equivalent for locally small categories. Proof. (4.1 4.2). For A C 0 and B D 0, define and η A = Φ A,B (id F A ) : A GF A ε B = Φ 1 A,B (id GB) : F GB B Check that these define components of natural transformations which satisfy the triangular identities. (4.2 4.1). Define and Φ A,B (F A f B) = A η A GF A Gf GB Ψ A,B (A g GB) = F A F g F GB ε B B Check that these maps are mutually inverse and natural in A and in B. Some examples of adjunctions are as follows: ˆ Let P and Q be posets. Recall that a functor F : P Q is precisely an order-preserving map. Thus a pair of functors P F Q form an adjoint pair F G if and only if (F, G) is a Galois connection, i.e. if for all p P and q Q we have G F (p) Q q if and only if p Q G(q) ˆ The forgetful functor U : Gp Set has a left-adjoint F : Set Gp, which assigns to a set X the free group F X generated by X. This is so because homomorphisms F X G correspond naturally with functions X U G. ˆ The forgetful functor U : Top Set has both a left-adjoint D : Set Top, which endows a set with the discrete topology, and a right-adjoint T : Set Top, which endows a set with the trivial topology. ˆ Let KHaus be the category of compact Hausdorff spaces and continuous maps. The inclusion functor I : KHaus Top has a left-adjoint β : Top KHaus, which is characterised as follows: given a continuous map f : X K with K compact Hausdorff there is a unique map βf : βx K such that X f K = X η X βx βf K where η X is the unit of the adjunction. Thus βx is the Stone-Čech compactification of X. ˆ Given a set A, the functor ( ) A : Set Set has a right-adjoint ( ) A : Set Set. The correspondence between maps B A C and B C A is given by currying and uncurrying; for instance, f : B A C corresponds with f : B C A, where f(b)(a) = f(b, a).e 14
4.1 Preservation of limits There are many reasons why knowing that two functors form an adjoint pair is useful. The following result, known colloquially as RAPL, is something which comes up all over the place and is important to know. Theorem 4.4 (Right-adjoints preserve limits). Suppose G : D C is a functor with a left-adjoint F : C D. Let d : J D be a diagram in D and suppose lim(d) exists. Then Gd : J C has a limit in C, and lim(gd) = G(lim(d)). Proof. Write (L, λ i ) = lim(d) and let µ i : F L d(i) be the map corresponding with λ i under the adjunction. Let (C, ε i ) be a cone over Gd in C. Thus for each i J 0 we have a C-morphism ε i : C Gd(i). Then each ε i corresponds under the adjunction with some η i : F C d(i) in D, and by naturality (F C, η i ) forms a cone over d in D. Hence there is a unique morphism u : F C L such that λ i u = η i. But then u corresponds under the adjunction with a unique map v : C GL satisfying µ i v = ε i. So (GL, µ i ) is a limit of Gd in C. Corollary 4.5 (Left-adjoints preserve colimits). Suppose F : C D is a functor with a rightadjoint G : D C. Let d : J C be a diagram in C and suppose colim(d) exists. Then F d : J D has a limit in D, and colim(f d) = F (colim(d)). Some examples are as follows: ˆ We saw above that there is a chain of adjunctions D U T, where U : Top Set is the forgetful functor and D, T : Set Top are the functors endowing a set with the discrete and trivial topologies, respectively. Thus, for example, we now know: Disjoint unions and quotients of discrete spaces are discrete. Products and intersections of trivial spaces are trivial. The underlying set of a product (or disjoint union, or quotient, or intersection, or...) of topological spaces is the product (or disjoint union or...) of their underlying sets. ˆ The adjunction β I tells us that any limit (e.g. product, intersection) of compact Hausdorff spaces is compact Hausdorff. It may not be true that a colimit of compact Hausdorff spaces is compact Hausdorff; for instance, the singleton space 1 is compact Hausdorff, but the disjoint union 1 = N (with the discrete topology) n N is not compact. This is a coproduct in Top because it is so in Set and as a left-adjoint the functor D : Set Top preserves colimits. The coproduct of N-many 1s in KHaus does exist, though: it s the Stone-Čech compatificaton βn of N. 15
Theorem 4.6. If d : J Top is a diagram, with J small, let and U : Top Set be a forgetful functor. Then lim(d) and colim(d) both exist, and moreover U(lim(d)) = lim(u d) and U(colim(d)) = colim(u d) That is, the underlying set of a limit or colimit in Top is precisely the limit or colimit of the underlying sets. Proof. U is both a left-adjoint and a right-adjoint since we have the chain of adjunctions D U T, so U preserves both limits and colimits. Thus if a limit or colimit of d exists then its underlying set must be the limit or colimit in Set of Ud. Moreover, these limits and colimits really do exist: they exist in Set since Set has all small limits and colimits (as can readily be seen by Theorem 3.7). We can then endow lim(d) with the coarsest topology making all the cone functions continuous; and colim(d) with the finest topology making all the cocone functions continuous. Proposition 4.7. A category C has limits (resp. colimits) of shape J if and only if the constant functor functor : C [J, C] has a right-adjoint (resp. left-adjoint). colim lim Proof. having a right-adjoint lim would mean that for all C-objects C and all diagrams d : J C there is a natural correspondence C ε d C u lim(d) where C : J C is the constant diagram with value C on objects and id C on morphisms. But this is precisely the assertion that if (C, ε i ) is a cone over d then there is a unique morphism u : C lim(d), and u factors appropriately by naturality of the adjoint correspondence. 4.2 Kan extensions Given any functor p : C C and a category C, there is a functor p : [C, D] [C, D] defined by p (C F p D) = F p : C C F D Definition 4.8. Let p : C C and F : C D be functors. If p : [C, D] [C, D] has a left-adjoint p!, then we say Lan p (F ) = p! (F ) : C D is the left Kan extension of F along p. If p has a right-adjoint p, then we way Ran p (F ) = p (F ) : C D is the right Kan extension of F along p. 16
For example, suppose C = 1, the terminal category (with one object and only an identity morphism), that! : J 1 is the unique morphism and that d : J C is a diagram of shape J in C. Then Lan! (d) = lim(d) : 1 C is the (functor picking out the) limit of d in C. Likewise Ran! (d) = colim(d) : 1 C is the (functor picking out the) colimit of d in C. References [1] Steve Awodey. Category Theory. Oxford Logic Guides. Oxford University Press, 2010. [2] Julia Goedecke. Part III Category Theory. 2011. [3] Saunders Mac Lane. Categories for the Working Mathematician. Categories for the Working Mathematician. Springer, 1998. 17