LIMITS IN CATEGORY THEORY


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1 LIMITS IN CATEGORY THEORY SCOTT MESSICK Abstract. I will start assming no knowledge o category theory and introdce all concepts necessary to embark on a discssion o limits. I will conclde with two big theorems: that a category with prodcts and eqalizers is complete, and that limits in any category can be redced to limits o Homsets by means o a natral transormation. Contents 1. Categories 1 2. Fnctors and Natral Transormations 3 3. Limits 4 4. Pllbacks 6 5. Complete Categories 8 6. Another Limit Theorem 9 Reerences Categories Category theory is a scheme or dealing with mathematical strctres in a highly abstract and general way. The basic element o category theory is a category. Deinition 1.1. A category C consists o three components: (1) A collection 1 o objects Ob(C ). Instead o C Ob(C ), we may write simply C C. (2) A collection o morphisms Ar(C ), and with each morphism, two associated objects, called the domain dom and the codomain cod. The set o morphisms with domain A and codomain B is written Hom C (A, B) or simply C (A, B) and called a Homset. A morphism can be thoght o as an arrow going rom its domain to its codomain. Indeed, I will se the words morphism and arrow interchangeably. Instead o C (A, B), : A B may be written, where A and B are already nderstood to be objects in C. (3) A composition law, i.e., or every pair o Homsets C (A, B) and C (B, C), a binary operation : C (A, B) C (B, C) C (A, C). Instead o (, g) we write g or g. Composition mst satisy the ollowing two axioms. Date: Agst 17, For the scope o this paper, I will not attempt to make the word collection precise. Note, however, that it is oten too big to be a set. 1
2 2 SCOTT MESSICK Category Objects Morphisms Set sets nctions Top topological spaces continos maps Grp grops homomorphisms o grops Any poset elements o the set exactly one arrow or every Table 1. Examples o categories (a) Associativity. I C (A, B), g C (B, C), and h C (C, D), then h (g ) = (h g) = h g = hg. (b) Identities. For every object C C there exists an identity arrow 1 C C (C, C) sch that or every morphism g C (A, C) and h C (C, B), 1 C g = g and h 1 C = h hold. The qintessential example o a category is the category o sets, Set. The objects are all sets, and the morphisms all nctions between sets (with the sal composition o nctions). The categories o topological spaces and grops are similar; in act, there is a category like this or almost every branch o mathematics; the objects are the strctres being stdied, and the morphisms are the strctrepreserving maps. The two axioms or composition o morphisms can be restated diagramatically as ollows. For every object C, there exists an identity arrow 1 C sch that the ollowing diagram commtes or every g, h: A g C h 1 C 1 C 1 C g C h B Given objects A, B, C, D and morphisms between them, the ollowing diagram always commtes: A B h g g g C h D A commtative diagram is one where, between any two given objects, composition along every (directed) path o arrows yields the same morphism between those objects. These diagrams also exempliy the sal way o illstrating concepts o category theory, representing objects as nodes and morphisms as arrows between them. The ollowing concept is important. Deinition 1.2. A morphism : A B is an isomorphism i there exists a morphism g : B A sch that g = 1 B and g = 1 A. Then the objects A, B are said to be isomorphic. Loosely speaking, isomorphic objects look the same in a category, becase arrows to or rom one can be niqely mapped throgh the isomorphism to arrows
3 LIMITS IN CATEGORY THEORY 3 to or rom the other. In terms o categorical strctre, thereore, they are indistingishable. For example, two sets o the same cardinality are isomorphic in Set, and are indeed the same thing i nctions between sets are all that is being considered. There are also generalizations o srjective and injective arrows, bt they are not needed here. 2. Fnctors and Natral Transormations Categories, in part, embody the idea that any notion o a mathematical object shold come with a notion o maps between two sch objects. Sets come with nctions, grops with homomorphisms, topological spaces with continos maps, and so on. Similarly, categories come with nctors. Deinition 2.1. A nctor F : C D is a map which associates with every object C C and object F (C) D, and with every morphism C (C 1, C 2 ) a morphism F () DF (C 1 ), F (C 2 ), and which preserves composition and identities, as in: F (1 C ) = 1 F (C) holds or every object C C. Whenever h, g, are arrows in C sch that h = g, it also holds that F (h) = F (g) F (). Parentheses may be omitted, as in C F C and F. There are many simple examples o nctors, orgetl nctors rom Grp or Top to Set which take objects to their nderlying sets, ree nctors going the other way (e.g. ptting the trivial topology on every set), nctors rom little categories that pick ot diagrams in their codomain (which will be important later), and so on. One important kind o nctor is given by Homsets. Observe that i C is any category, with any object C, then Hom C (C, ) gives a nctor H : C Set. This nctor maps arrows by letcomposition, i.e., given : A B, H : Hom C (C, A) Hom C (C, B) is deined by (H)(g) = g. A natral transormation is, in trn, a morphism o nctors. Given nctors F, G : C D, one may imagine the image o F as a bnch o objects sitting in D, with some arrows between them highlighted. Similarly one may imagine the image o G as another bnch o objects. Loosely speaking, a natral transormation will be a way o getting rom the irst pictre to the second pictre, sing the arrows o D, in a natral way, i.e., in the same way or every object. Deinition 2.2. Given nctors F, G : C D, a natral transormation η : F G is a collection o arrows in D, speciically, one arrow or each object X o C, called 2 η X, sch that the ollowing diagram commtes or every X, Y C. X F X F F Y η X Y GX G GY Deinition 2.3. A natral isomorphism is a natral transormation in which every arrow is an isomorphism. η Y 2 I m switching notation slightly here, to avoid nested sbscripts.
4 4 SCOTT MESSICK 3. Limits The notion o a limit in category theory generalizes varios types o niversal constrctions that occr in diverse areas o mathematics. It can show very precisely how thematically similar constrctions o dierent types o objects, sch as the prodct o sets or grops o topological spaces, are instances o the same categorical constrct. Consider the Cartesian prodct in sets. X Y is sally deined by internally constrcting the set o ordered pairs {(x, y) x X and y Y }. Bt it can also by identiied as the set which projects down to X and Y in a niversal way, that is to say, doing something to X Y is the same as separately doing something to X and Y. Deinition 3.1. A prodct o objects A, B in a category C is an object, C, together with morphisms p : C A and q : C B, called the projections, with the ollowing niversal property 3. For any other object D C with morphisms : D A and g : D B, there is a niqe morphism : D C sch that p = and q = g. In other words, every (D,, g) actors niqely throgh (C, ). D g C q B p A Example 3.2. In the Set, the prodct is the Cartesian prodct. The projections p : A B A and q : A B B are given by p(a, b) = a and q(a, b) = q. Given another set D with arrows : D A and g : D B, the niqe arrow : D A B is given by (d) = ((d), g(d)). That this commtes and is the only arrow doing so are transparent, as or example, (p )(d) = p((d)) = p((d), g(d)) = (d). Example 3.3. In Grp, the prodct is the direct prodct o grops. The constrction is similar to set; the prodct is given by the nderlying Cartesian prodct, with a grop operation constrcted elementwise rom those o the actors. To demonstrate that the niversal arrow exists and is niqe, it sices to show that the nction given by the same constrction is in act a homomorphism, given that, becase we are in the category o grops, and g are also homomorphisms. (Trivially, the projections are homomorphisms.) This proo is straightorward: (3.4) (d 1 d 2 ) = ((d 1 d 2 ), g(d 1 d 2 )) = ((d 1 )(d 2 ), g(d 1 )g(d 2 )) = ((d 1 ), g(d 1 ))((d 2 ), g(d 2 )) = (d 1 )(d 2 ) Example 3.5. In Top, the prodct is the sal prodct o topological spaces. In act, this prodct is oten deined as the coarsest topology which makes the projections continos, which is exactly what is needed to make the analogos constrction work. paper. 3 A ormal deinition o niversal properties exists, bt is nnecessary or the prposes o this
5 LIMITS IN CATEGORY THEORY 5 Example 3.6. In a poset, the prodct is the greatest lower bond, i it exists. This provides not only an example which is very dierent rom sets, bt also one showing that the prodct doesn t always exist. Let c = glb(a, b). There is only one choice o projections. I d has arrows to a and b, it means a d and b d, so d is an pper bond. Bt c is a least pper bond, so c d. This gives the niversal arrow which easily commtes and is niqe becase arrows are scant in this category. Proposition 3.7. The prodct o any two objects in a category, i it exists, is niqe p to niqe isomorphism. Proo. Let A and B be objects in a category, and C, p C, q C and D, p D, q D be prodcts o A and B. By the niversal property, there exist niqe morphisms : C D and g : D C which commte with the projections. This gives p C g = p D and p D = p C and hence p C g = p C. Similarly, q C g = q C. Ths, g is an arrow rom C to itsel which commtes with the projections. Bt by the niversal property there can only be one morphism rom C to itsel which commtes with the projections, and the identity sices. Hence g = 1 C. The other way arond is similar. C A g B D C A 1 C g B C Remark 3.8. A prodct can be generalized in an obvios way to any nmber o actors other than two. Later I will speak o a category with all small prodcts ; this jst means the prodcts o any set o objects exists in the category. In Set it s clear that all small prodcts exist. The second most important example o a limit is an eqalizer. In sets, and in many similar categories, this is jst the sbset o the domain o two parallel arrows where those two nctions are eqal. Deinition 3.9. An eqalizer o two arrows, g : X Y in a category C is an object, E, together with a morphism e : E X sch that e = g e, with the ollowing niversal property: or any O C with a morphism m : O X sch that m = g m, there is a niqe morphism : O E sch that m = e. This eqalizer may be denoted eq(x, Y ).
6 6 SCOTT MESSICK E e X Y g m O Remark In Grp and Top, eqalizers are constrcted exactly the same way. In Grp, it can be viewed as a dierence kernel. (In act, kernels can also be viewed as limits). Now we re ready or the general notion o a limit, bt irst, it s sel to deine a cone. Notice that in the prodct and the eqalizer, the projections played the same role as the morphism e. In what ollows, J shold be thoght o as a small category, sch as two discrete objects (no morphisms except identities) in the case o prodcts, or a jst a pair o objects with a pair o arrows in the case o eqalizers. The nctor F : J C shold be thoght o as a diagram o that shape in the category C. Ths the limit o the diagram is taken. Deinition Given a nctor F : J C, a cone o F is an object N C together with morphisms ψ X : N F (X) or every X J sch that or every morphism : X Y in J, the triangle commtes, i.e. F ψ X = ψ Y. ψ X ψ Y N ψ Z F (X) F (Y ) F (Z) F () F (g) Deinition A limit o a nctor F : J C is a niversal cone 4 L, φ X. That is, or every cone N, ψ X o F, there is a niqe morphism : N L sch that φ X = ψ X or every X J. The limit object may be written lim i J F (i). N ψ X ψ L Y φ X φ Y F (X) F (Y ) F () Proposition The limit o any diagram in a category, i it exists, is niqe p to niqe isomorphism. 4. Pllbacks The pllback (iber prodct) is the last limit I will deine explicitly. There are many other important limits, bt pllbacks will be my example or how all limits come rom prodcts and eqalizers. 4 The term limit is overloaded to mean either the cone, i.e., the object with the arrows, or jst the object.
7 LIMITS IN CATEGORY THEORY 7 Deinition 4.1. A pllback is a limit o a diagram o the ollowing orm: A C B. That is, a pllback is an object D with morphisms p 1 : D A and p 2 : D B which make the sqare commte and are niversal, i.e. or every other object Q with morphisms q 1 : Q A and q 2 : Q B, there is a niqe morphism : Q D which makes the diagram commte. The pllback, interpreted as the object D, may be written A C B. Q D B A Proposition 4.2. In Set, the pllback is given by the set X Z Y = {(x, y) (x) = g(y)} where : X Z and g : Y Z, together with the restricted projection maps p 1, p 2 into X and Y. Proo. Let D, q 1, q 2 be another cone o the same diagram. We have (q 1 (d)) = g(q 2 (d)) or every d D. Ths the pairs (q 1 (d), q 2 (d)) are in X Z Y. Let (d) = (q 1 (d), q 2 (d)). Proposition 4.3. In any category, a pllback can be constrcted sing a prodct and an eqalizer. C A B p2 B p 1 A p 1 The prodct A B indces two parallel diagonal arrows to C, o which the eqalizer can be taken. gp 2 C g E p 2e p 1e e A B p2 B p 1 A Now, consider any other cone commting with the pllback diagram. The niversal property o the prodct gives a niqe arrow to the prodct. By a diagram chasing argment, this arrow can be seen to eqalize the diagonal arrows, which gives a niqe arrow to the eqalizer that makes the diagram commte. Ths, the eqalizer is the pllback. p 1 gp 2 C g
8 8 SCOTT MESSICK 5. Complete Categories Deinition 5.1. A category is complete i every small 5 diagram has a limit. Theorem 5.2. Let F : J Set be any small diagram in Set. Then the limit o F is the set L = lim i J F (i) = {(x i ) i J F (i) (F )(x i ) = x cod Ar(J )} Remark 5.3. An eqivalent condition to the one given is that (F )(x i ) = (F g)(x j ) whenever, g Ar(J ) and cod = cod g. For g can be the identity, which gives the statement in the theorem, and to go the other way, note that i cod = cod g = k then (F )(x i ) = x k = (F g)(x j ). This makes it clear that an eqalizer is being taken in order to prodce the limit. Proo. The limit cone is L with restricted projection maps (p i ). Let (N, (ψ i ) i J ) be any other cone. We have (F )(ψ i (n)) = (F g)(ψ j (n)) whenever cod = cod g, or every n N. Ths the tples (ψ i (n)) i J are in L. Let (n) = (ψ i (n)) i J. Theorem 5.4. A category with all eqalizers and all small prodcts is complete. Proo. This proo is a carel and slightly clever generalization o the idea in the previos proo. Actal eqality no longer exists, so an eqalizer has to be sed to do the trick. We will take two prodcts and ind two arrows between them o which to take the eqalizer, which will be the limit. The prodct is taken irst over all objects in the diagram, then over the codomains o all arrows in the diagram, indexed by arrows. F (cod ) i F (i) F (dom ) v F p F (cod ) p F (cod ) The irst prodct with its projections (repeated i necessary) can make a cone or the second prodct in two dierent ways, as shown in the triangle on top and the sqare on the bottom o the irst igre. This gives the two niversal arrows and v as shown, which respectively make all the triangles commte, or all the sqares. 5 A diagram is small i the collection o objects is a set.
9 LIMITS IN CATEGORY THEORY 9 F (i) F (cod ) φ i E e p i i F (i) F (dom ) v F p F (cod ) p F (cod ) The eqalizer is taken as shown, and the arrows φ i = p i e are ormed to make the cone E, φ i. That it s a cone can be seen by inspecting the diagram. Given : j k, we have e = ve, p e = p ve, p k e = F p j e, phi k = F phi j. Then any other cone Q, (psi i ), gives a niqe map t to the prodct, by its niversal property, bt since it s a cone, t = vt, so there is a niqe arrow s : Q E by the niversal property o the eqalizer. 6. Another Limit Theorem The ollowing is a beatil and ascinating theorem, and in a paper o larger scope, mch more cold be done with it. Even withot the prpose o proving other theorems, it illstrates a lot o intition behind the notion o a categorical limit. Theorem 6.1. Let F : J C be a diagram in C. Then an object X C is a limit o F i and only i there is a natral isomorphism Hom C (C, X) = lim i J Hom C (C, F (i)) where the limit on the right is in Set and hence exists. Proo or prodcts. First sppose that C is the prodct A B. We need to ind a bijection o sets, Hom(D, C) = Hom(D, A) Hom(D, B). Given arrows rom D to A and B gives s a cone o F, so since C is the prodct we have a niqe arrow given by the niversal property. For the other direction, given any arrow h : D C we simply take p h, q h, so we have a bijection. To show natrality, let α : M N be an arrow in C. Then the ollowing diagram mst commte: Hom(M, C) Hom(M, A) Hom(M, B) Hom(N, C) Hom(N, A) Hom(N, B) Here the vertical arrows are jst letcomposition by α and the horizontal arrows are jst elementwise rightcomposition by p, q. The diagram commtes by associativity. On the other hand sppose we have sch a natral isomorphism or some object C. We need to ind the projections in order to orm a prodct. To do this, we consider a particlar, enlightening case o the natral isomorphism: Hom(C, C) = Hom(C, A) Hom(C, B)
10 10 SCOTT MESSICK Even in an arbitrary category, we know Hom(C, C) has an identity element. By plgging the identity throgh the isomorphism, we get p : C A and q : C B. Now let D,, g be another cone. We can prove the niversal property by sing the ollowing commtative diagram: Hom(D, C) Hom(D, A) Hom(D, B) Hom(C, C) Hom(C, A) Hom(C, B) (, g) 1 C (p, q) We know we can constrct the vertical arrows becase the bijection gives s : D C rom (, g). The vertical arrows are letcomposition by. The commtative diagram then states that p = and q = g. Notice that this theorem illstrates what a prodct morally is: an object sch that speciying a map to the object is the same as speciying a map to the actors. General proo. The general case proceeds similarly. Let L be the limit o F : J C. We need to ind a bijection o sets, Hom(X, L) = lim Hom(X, F (i)). What is an element o the limit on the righthand side? Well, we have an explicit description o a limit in Set. Bt we have to be carel, becase the nctor here is not F itsel bt HF, where H is the Homnctor Hom(X, ). Then the ollowing is tre or an element (ψ i ) o the limit: (HF )(ψ i ) = ψ j whenever : i j in J. Bt we know what H does to arrows, so we have F ψ i = ψ j wheenver : i J in J. Ths, the statement that a tple o arrows (ψ i ) is in the limit o Homsets is exactly the statement that X, (ψ i ) is a cone o F. This gives a niqe arrow : X L and everything proceeds jst as beore. To go the other way, we again take the bijection Hom(L, L) = lim Hom(L, F (i)) and eed the identity throgh it, yielding a cone L, (φ i ) as beore, which we mst prove is niversal. Let X, (ψ i ) be any other cone. Hom(X, L) lim Hom(X, F (i)) Hom(L, L) lim Hom(L, F (i)) (ψ i ) 1 L Jst as beore, we can ind with the bijection and constrct the commtative diagram to complete the proo. (φ i )
11 LIMITS IN CATEGORY THEORY 11 Reerences [1] Mac Lane, Sanders. Categories or the Working Mathematician. Second Edition, [2] Gillo, Bert and Haris Skiadas. WOMP 2004: Category Theory. [3] Anders, Alan. DRP Notes, Winter 2007.
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