Locally Presentable Categories and Localizations

5 November 2018

1.1 Some conventions and terminology A A of a 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 ∈ C instead of X ∈ 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 F : D → C where D is a small category. The values of a diagram F : D → C are the objects F d ∈ C with d ∈ D, which are sometimes denoted by Fd instead. Hence the following expressions mean the same thing —namely, an object C ∈ C together with a universal cocone {fd : Fd → C}d∈D in case that it exists:

colim F ; colim F ; colim Fd. D d∈D

1.2 Canonical diagrams Let A be a small full subcategory of a category C. For an object X ∈ C, the canonical diagram for X with respect to A is the

U :(A ↓ X) −→ C (1.1) defined as U(f : A → X) = A. The objects of the comma category (A ↓ X) are the f : A → X with A ∈ A, and a 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 ∈ C is a canonical colimit of objects of A if the cocone {f : U(f : A → X) → X} for all A ∈ A and f ∈ 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 ∈ 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 ∈ N. Quite often we consider dense 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 ∈ D there is an object d ∈ 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 ∈ D 0 and a morphism f : d2 → d with f ◦ g = f ◦ g . Every category with finite colimits is filtered, since the existence of ensures that (i) holds and the existence of 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) d∈D d∈D 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 ∈ 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 ∈ 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 ∈ 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 | Ri where the set X of generators and the set R of relations are finite. The 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 . 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 ℵ0 is regular. Every successor cardinal is regular. Uncountable regular 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 {di}i∈I of morphisms in D with |I| < λ there is an object d ∈ D and morphisms fi : di → d for all i ∈ I.

(ii) For every set of parallel morphisms {gi : d1 → d2}i∈I in D with |I| < λ there is an object d ∈ D and a morphism f : d2 → d with f ◦ gi = f ◦ gj for all i, j ∈ I. A diagram F : D → C is λ-small if |D| < 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 |X| < 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 ∈ 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 ℵ0-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 ∈ 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 = colimd∈D Ad, then the corresponding cocone {fd : Ad → X}d∈D 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

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 ∈ 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. These names come from the fact that the morphism lLX : LX → LLX is an isomor- phism for all X and moreover lLX = LlX . Thus the monad (L, l) is idempotent. Objects in the essential image of L (which is also the closure of A under iso- morphisms) are called L-local objects, and morphisms f : X → Y in C such that Lf : LX → LY is an isomorphism are called L-equivalences. The L-local objects and the L-equivalences are orthogonal in the sense that there is a bijection

f ∗ : C(Y,W ) ∼= C(X,W ) (1.3) whenever f : X → Y is an L-equivalence and W is L-local. Moreover, the classes of L-local objects and L-equivalences determine each other by orthogonality. For every object X ∈ C, the unit lX : X → LX is an initial morphism from X into an L-local object and also a terminal L-equivalence out of X.

1.6 Orthogonality classes For a category C and a class M of morphisms in C, we denote by M⊥ the class of objects orthogonal to all morphisms in M; that is, those objects W such that (1.3) holds for all f : X → Y in M. Every such class M⊥ is called an orthogonality class. Orthogonality classes are closed under all limits that exist in C. If M is a set, then M⊥ is called a small-orthogonality class. If domains and codomains of morphisms in M are λ-presentable for some regular cardinal λ, then M⊥ is called a λ-orthogonality class. Thus every λ-orthogonality class in a locally λ-presentable category is a small-orthogonality class, and it is closed under λ-filtered colimits.

Theorem 1.11. Every small-orthogonality class in a locally presentable category is reflective.

Proof. One constructs explicity a reflection by means of an old well-known procedure called “small object argument”. Theorem 1.11 can be refined as follows.

4 Theorem 1.12. If C is a locally λ-presentable category, then the following assertions are equivalent for a full subcategory A:

(i) A is a λ-orthogonality class in C.

(ii) A is a reflective subcategory of C closed under λ-filtered colimits.

Moreover, each of these assertions implies that A is locally λ-presentable.

1.7 Restricted Yoneda embedding For a small full subcategory A of a category C, the restricted Yoneda functor is the functor Aop EA : C −→ Set (1.4) sending each object X ∈ C to the functor C(−,X): Aop → Set. Its basic properties are stated into the following theorem. We omit proofs.

Theorem 1.13. Let EA be the restricted Yoneda functor for a small full subcategory A of a category C.

(a) EA is fully faithful if and only if A is dense in C.

(b) For a regular cardinal λ, the functor EA preserves λ-filtered colimits if and only if every object of A is λ-presentable in C.

(c) If C is cocomplete, then EA has a left adjoint. Corollary 1.14. If C is locally λ-presentable and A is a set of representatives of all isomorphism classes of λ-presentable objects of C, then EA is an embedding into a op reflective full subcategory of SetA .

Proof. The functor EA is fully faithful because A is dense in C, and EA has a left adjoint since C is cocomplete.

1.8 The Representation Theorem The image of the restricted Yoneda functor (1.4) falls into the collection of functors Aop → Set that preserve all existing limits, since     C lim F,X = C colim F op,X ∼= lim C(F,X) D Dop D for every X ∈ C and every diagram F : D → Aop for which a limit exists in Aop (hence a colimit for F op : Dop → A exists in A). This remark opens the way to the next characterization of locally presentable categories. op Aop For a small category A, we denote by Contλ A the full subcategory of Set whose objects are functors F : Aop → Set that preserve all λ-small limits that exist in Aop. Such functors are called λ-continuous.

5 Theorem 1.15. For a regular cardinal λ and a category C, the following statements are equivalent:

(i) C is locally λ-presentable.

op (ii) C' Contλ A for some small category A.

op (iii) C is equivalent to a full reflective subcategory of SetA closed under filtered λ-colimits for some small category A.

Proof. Suppose first that C is locally λ-presentable and choose a set A of represen- tatives of isomorphism classes of all λ-presentable objects of C. Then the restricted Yoneda embedding Aop EA : C −→ Set Aop is fully faithful and its essential image EssIm EA is a reflective subcategory of Set by Corollary 1.14. Since representable functors preserve limits and Aop is closed under λ-small limits, we infer that

op EssIm EA ⊆ Contλ A . (1.5)

op Now every object F ∈ SetA is a canonical colimit of representable functors by the Density Theorem, and the canonical colimit is λ-filtered if F preserves λ-small op limits. Hence Contλ A is contained into EssIm EA, so (1.5) is an equality. Moreover, since EA preserves λ-filtered colimits by part (b) of Theorem 1.13 and the category C is the closure of A under λ-filtered colimits, the category EssIm EA Aop is the closure of the image of EA in Set under λ-filtered colimits, and this means that EssIm EA is closed under λ-filtered colimits. This proves that (i) implies (ii) and (iii). The implications (ii) ⇒ (iii) ⇒ (i) follow from Theorem 1.12.

Reference: J. Ad´amekand J. Rosick´y, Locally Presentable and Accessible Cate- gories, Cambridge University Press, Cambridge, 1994.

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