LIMITS Abstract. Limits and colimits. Applications. 1. Limits and colimits 1.1. Limits. 1.1.1. Functor categories. Definition 1.1. Let D and C be categories. ( ) ! D (a) The class of all functors F : D C is denoted by C 0. (b) The class( of all) natural transformations( ) τ : F ! G between D D objects of C 0 is denoted by C ( 1. ) ! D (c) Given an element F : D C of C 0, the identity natural ! transformation 1F has components (1F )x = 1xF : xF xF at each vertex x of D. (d) Given natural transformations( ) τ1 : F0 ! F1 and τ2 : F1 ! F2 D between objects of C 0, the composite natural transformation τ1τ2 : F0 ! F2 is defined by its components (τ1τ2)x = (τ1)x(τ2)x at each vertex(( x of) D(. ) ) D D D (e) Then C = C 0 ; C 1 is known as a functor category. Example 1.2. Let G be a group that is realized as a category G with a single object X, and a morphism g for each group element g (compare Example 1.4, Categories). Consider the category Z of abelian groups. Then the functor category ZG is the category of G-modules. Indeed, given a functor r : G ! Z with Xr = M, each morphism gr (for g 2 G) is an automorphism of the abelian group M, and the functoriality of r means that mgrhr = m(gh)r for m 2 M and g; h 2 G. Example 1.3. A poset is discrete if it has the form (I; =) for a set I. Then for 2 = f0; 1g construed as a discrete poset category, the functor 2 category C has an object class consisting of pairs (x0; x1) of objects of C, and a morphism class consisting of pairs (f0 : x0 ! y0; f1 : x1 ! y1) of morphisms of C. 1.1.2. Limits. Definition 1.4. Let J and C be categories, and let F : J ! C be a functor. 1 2 LIMITS (a) The diagonal functor ∆: C ! CJ sends each object X of C to the constant functor [X]: J ! C. Each morphism f : X ! Y of C is sent to the natural transformation [f]:[x] ! [y] with component [f]j = f at each object j of J. (b) Let [F ] be the constant functor [F ]: 1 ! C. Then the limit lim F of F is a terminal object of the comma category (∆ # [F ]). − The comma category (∆ # [F ]) in Definition 1.4(b) is built on the pair [F ] C −!∆ CJ − 1 of functors whose common codomain is the functor category CJ . • An object (Y; Y ∆ ! F; 1) of the comma category (∆ # [F ]) consists of a natural transformation f :[Y ] ! F , specified by its components fj : Y ! jF at each object j of J. • In particular, the terminal object lim F of the comma category − (∆ # [F ]) consists of an object of C itself, informally denoted as lim F , together with a natural transformation π : [lim F ] ! F , − − specified by its components π : lim F ! jF for each object j j − of J. These components are known as projections. • The terminality of lim F means that for each object Y of C with − morphisms fj : Y ! jF for j 2 J0, there is a unique comma category morphism (lim f; 1 ) 2 C(Y; lim F ) × 1(1; 1) − 1 − such that the diagrams [lim f] lim f − / − / (1.1) [Y ] [lim F ] or Y > lim F − >> − >> zz f π >> z > zz π fj > }zz j F / F jF 1F commute (in the second case, for each object j of J). (For the first diagram, compare (1.1) in Structural specifications). 1.1.3. Products and pullbacks as limits. Products of pairs of objects are implemented as limits in which the index category J is the discrete Qposet category 2 of Example 1.3. More generally, an arbitrary product i2I Xi is implemented as a limit in which the index category J is the discrete poset category I given by the set I. LIMITS 3 Example 1.5. Let J be the join semilattice poset category with Hasse diagram 0 −! 0+1 − 1. Then for a functor F : J ! C with jF = Xj for j 2 J , the limit lim F is the pullback X π−0 X × X −!π1 X . 0 − 0 0 X0+1 1 1 × ! ! F Note that π0+1 : X0 X0+1 X1 X0+1 is the composite π0(0 0+1) = F π1(1 ! 0 + 1) . The diagram 8 XO 1 F f FF 1 FF(1!0+1)F FF π1 FF F# / / Y lim F X0+1 lim f − π0+1 ; − xx xx π0 xx xx (0!0+1)F f0 & x X0 summarizes this case of the right-hand diagrams in (1.1). 1.1.4. Equalizers. Let k or parallel denote the category ( 8 0 6 1 f with a parallel pair of non-identity arrows. Thus a functor F : k ! C f0 / corresponds to a parallel pair of arrows X0 / X1 in the category f1 C. The limit lim F is a morphism e: E ! X such that − 0 D AA AA lim d AAd − A A f0 / / / E e X0 X1 f1 commutes, i.e., such that ef0 = ef1, and such that d: D ! X0 with df = df implies the existence of a unique morphism lim d: D ! E 0 1 − with (lim d)e = d. The limit E is known as the equalizer of f and f . − 0 1 In the category of sets, or concrete categories of algebras, one may take e as the insertion of the subset E = fx 2 X0 j xf0 = xf1g into X0. 1.2. Colimits. Colimits are the duals of limits. Definition 1.6. Let J and C be categories, and let F : J ! C be a functor, with [F ] as the constant functor [F ]: 1 ! C. Then the colimit lim F of F is an initial object of the comma category ([F ] # ∆). −! 4 LIMITS The comma category ([F ] # ∆) in Definition 1.6 is built on the pair [F ] 1 −! CJ ∆− C of functors whose common codomain is the functor category CJ . • An object (1;F ! Y ∆;Y ) of the comma category ([F ] # ∆) consists of a natural transformation f : F ! [Y ], specified by its components fj : jF ! Y at each object j of J. • In particular, the initial object lim F of the comma category −! ([F ] # ∆) consists of an object of C itself, informally denoted as lim F , together with a natural transformation ι: F ! [lim F ], −! −! specified by its components ι : jF ! lim F for each object j of j −! J. These components are known as insertions. • The initiality of lim F means that for each object Y of C with −! morphisms fj : jF ! Y for j 2 J0, there is a unique comma category morphism (1 ; f) 2 1(1; 1) × C(lim F; Y ) 1 −! such that the diagrams 1 (1.2) F F / F or jF z >> zz > ιj zz >>fj ι f zz >> }zz > [lim F ] / [Y ] lim F / Y −! [lim f] −! lim f −! −! commute (in the second case, for each object j of J). Remark 1.7. Limits are also known as \projective limits" (because they come with projections) or \inverse limits," and may be written as \lim." Colimits, which may be written as \colim," are also known as \inductive limits" or \direct limits." Remark 1.8. As a mnemonic for the direction of the arrow under limit π and colimit symbols, one may think of the projections jF −j lim F − ι coming out of limits and the insertions jF −!j lim F going in to colimits. −! 1.2.1. Coproducts. Coproducts of pairs of objects are just colimits in which the index category J is the discrete poset category 2 `considered Pin Example 1.3. More generally, an arbitrary coproduct i2I Xi or i2I Xi is a colimit in which the index category J is the discrete poset category I given by the set I. LIMITS 5 ` Example 1.9. (a) If I is empty, the coproduct i2I Xi is just an initial object. f j 2 g R (b) Let X = xi i I be a subset of thatP is bounded above. Then R ≤ in the poset category ( ; ), the coproduct j2I xj is the supremum sup X. 2 (c) If thereP is an object X such that Xi = X for all i I, then the coproduct i2I Xi is the I-th multiple IX. 1.2.2. Pushouts. Definition 1.10. Let J be the meet semilattice poset category with Hasse diagram 0 − 0 · 1 −! 1. Then for a functor F : J ! C with jF = Xj for j 2 J0, the pushout −!ι0 ι−1 (1.3) X0 X0 +X0·1 X1 X1 is the colimit lim F . −! F F In the pushout (1.3), the composite (0 · 1 ! 0) ι0 = (0 · 1 ! 1) ι1 ! is the insertion ι0·1 : X0·1 X0 +X0·1 X1. The diagram 4 X1 (0·1!1)F f1 ι1 / / X0·1 X0 +X · X1 BY ι0+1 O 0 1 lim f −! ι0 · ! (0 1 0)F * f0 X0 summarizes this case of the right-hand diagrams in (1.2). 1.2.3. Coequalizers. For a functor F : k ! C yielding a parallel pair of f0 / arrows X / X in a category C, the colimit lim F is known as a 1 0 −! f1 coequalizer. It is a morphism q : X0 ! Q such that }> PO p }} } lim p }} −! }} f0 / / / X1 X0 q Q f1 commutes, i.e., such that f0q = f1q, and such that p: X0 ! P with f p = f p implies the existence of a unique morphism lim p: Q ! P 0 1 −! with q(lim p) = p.