Locally Presentable Categories and Localizations 5 November 2018 1.1 Some conventions and terminology A subcategory A of a category C is full if A(A; B) = C(A; B) for all objects A and B of A. It is common practice to use the same notation for a collection of objects of a category C and the full subcategory of C with those objects. It is also usual to write X 2 C instead of X 2 Ob(C). A category is small if the collection of its objects is a set, and essentially small if the isomorphism classes of its objects form a set. A diagram in a category C is a functor F : D!C where D is a small category. The values of a diagram F : D!C are the objects F d 2 C with d 2 D, which are sometimes denoted by Fd instead. Hence the following expressions mean the same thing |namely, an object C 2 C together with a universal cocone ffd : Fd ! Cgd2D in case that it exists: colim F ; colim F ; colim Fd: D d2D 1.2 Canonical diagrams Let A be a small full subcategory of a category C. For an object X 2 C, the canonical diagram for X with respect to A is the forgetful functor U :(A# X) −! C (1.1) defined as U(f : A ! X) = A. The objects of the comma category (A# X) are the morphisms f : A ! X with A 2 A, and a morphism in (A# X) from f : A ! X to f 0 : A0 ! X is a morphism g : A ! A0 such that f 0 ◦ g = f. We say that an object X 2 C is a canonical colimit of objects of A if the cocone ff : U(f : A ! X) ! Xg for all A 2 A and f 2 C(A; X) is universal, which implies that colim U ∼= X; (1.2) (A# X) where U is the canonical diagram (1.1) for X with respect to A. Note that (1.2) holds for all X if A = C, since id: X ! X is a terminal object in (C# X). Definition 1.1. A small full subcategory A ⊆ C is called dense if every object X 2 C is a canonical colimit of objects of A. For example, N is dense in the category Set of sets, since every set X is the union of all its finite subsets and hence it is the colimit of the diagram of all maps n ! X with n 2 N. Quite often we consider dense subcategories that are not small but essentially small. A colimit indexed by an essentially small category A is meant to be a colimit indexed by any set of representatives of isomorphism classes of objects of A. 1 1.3 Finitely presentable objects A category D is filtered if the following conditions hold: (i) For all d1; d2 2 D there is an object d 2 D and there are morphisms f1 : d1 ! d and f2 : d2 ! d. 0 (ii) For every two morphisms g : d1 ! d2 and g : d1 ! d2 there is an object d 2 D 0 and a morphism f : d2 ! d with f ◦ g = f ◦ g . Every category with finite colimits is filtered, since the existence of coproducts ensures that (i) holds and the existence of coequalizers ensures that (ii) holds. A diagram F : D!C is filtered if the category D is filtered. A colimit is filtered if it is indexed by a filtered category. An object X of a category C is said to be finitely presentable if the functor C(X; −): C! Set preserves filtered colimits, that is, ∼ C X; colim Fd = colim C(X; Fd) d2D d2D for every filtered diagram F : D!C for which colimD F exists. This implies that every morphism X ! colimD F factors through Fd for some d 2 D, and also that if two morphisms f1 : X ! Fd1 and f2 : X ! Fd2 compose to the same morphism X ! colimD F , then there is an object d 2 D with morphisms d1 ! d and d2 ! d such that f1 and f2 compose to the same morphism X ! Fd. Every finite colimit of finitely presentable objects is finitely presentable. Definition 1.2. A category is C is locally finitely presentable if (a) C is cocomplete; (b) there is only a set of isomorphism classes of finitely presentable objects in C; (c) every object X 2 C is a filtered colimit of finitely presentable objects. Example 1.3. A set X is finitely presentable in Set if and only if X is finite. The category Set is locally finitely presentable since every set is a filtered union of its finite subsets. Example 1.4. A group G is finitely presentable if it admits a presentation hX j Ri where the set X of generators and the set R of relations are finite. The category of groups is locally finitely presentable since every group is a filtered union of its finitely generated subgroups. The class of finitely presentable groups is the closure under finite colimits of the class of finitely generated free groups. Example 1.5. If A is any small category, then A(−;A) is a finitely presentable op object of SetA by the Yoneda Lemma: op op SetA (A(−;A); colim F ) ∼= (colim F )A = colim(FA) ∼= colim SetA (A(−;A);F ): By the Density Theorem, every functor Aop ! Set is a canonical colimit of repre- sentable functors. However, this colimit is not filtered in general. Thus, let A¯ be op the closure under finite colimits of the set of representable objects in SetA . Then every object of A¯ is still finitely presentable. Moreover, A¯ is now filtered and every op op object of SetA is a canonical colimit of objects of A¯. Therefore SetA is locally finitely presentable. 2 1.4 Locally presentable categories A cardinal λ is regular if it is infinite and it is not a sum of a smaller number of smaller cardinals. The first infinite cardinal @0 is regular. Every successor cardinal is regular. Uncountable regular limit cardinals are called weakly inaccessible and their existence cannot be proved in ZFC; in fact, the claim that weakly inaccessible cardinals exist is the smallest instance of a hierarchy of large-cardinal axioms. Definition 1.6. For a regular cardinal λ, a category D is λ-filtered if the next conditions hold: (i) For every set fdigi2I of morphisms in D with jIj < λ there is an object d 2 D and morphisms fi : di ! d for all i 2 I. (ii) For every set of parallel morphisms fgi : d1 ! d2gi2I in D with jIj < λ there is an object d 2 D and a morphism f : d2 ! d with f ◦ gi = f ◦ gj for all i; j 2 I. A diagram F : D!C is λ-small if jDj < λ, where the cardinality of a small category is defined as the cardinality of its set of morphisms. Every category closed under λ-small colimits is λ-filtered. If µ > λ, then every µ-filtered category is λ-filtered. A diagram F : D!C is λ-filtered if the category D is λ-filtered. A colimit is λ-filtered if it is indexed by a λ-filtered category. Definition 1.7. An object X of a category C is λ-presentable if C(X; −): C! Set preserves λ-filtered colimits. For example, a set X is λ-presentable in Set if and only if jXj < λ. If µ > λ, then every λ-presentable object is µ-presentable. Every colimit of a λ-small diagram of λ-presentable objects is λ-presentable. Definition 1.8. A category is C is locally λ-presentable if (a) C is cocomplete; (b) there is only a set of isomorphism classes of λ-presentable objects in C; (c) every object X 2 C is a λ-filtered colimit of λ-presentable objects. Definition 1.9. A category C is locally presentable if C is locally λ-presentable for some regular cardinal λ. The locally finitely presentable categories are the locally @0-presentable ones. Proposition 1.10. If C is a locally λ-presentable category, then the essentially small class A of all the λ-presentable objects in C is dense, and the canonical diagram for every object X 2 C with respect to A is λ-filtered. Proof. The category (A# X) is λ-filtered for each X since every λ-small colimit of λ-presentable objects is λ-presentable. Since C is locally λ-presentable, every object ∼ X is a λ-filtered colimit of objects of A. If X = colimd2D Ad, then the corresponding cocone ffd : Ad ! Xgd2D is cofinal in the canonical diagram for X with respect to A and hence the latter also has X as its colimit. 3 1.5 Reflective subcategories A full subcategory A of a category C is called reflective if the inclusion J : A ,!C has a left adjoint R: C!A. In this case, the composite L = JR defines a monad on C. The bijection C(X; JA) ∼= A(RX; A) = C(LX; JA) (where the second equality comes from the fact that A is full) means that for every morphism f : X ! A in C where A 2 A there is a unique morphism g : LX ! A such that g ◦ lX = f, where lX : X ! LX is the unit of the adjunction, that is, the morphism corresponding to the identity of RX under the bijection C(X; LX) ∼= A(RX; RX): The left adjoint R is called a reflection onto A, and L is called a localization.