nLab
locally presentable category

Contents

Idea

A locally presentable category is a category which contains a small set SS of small objects such that every object is a colimit over objects in this set.

This says equivalently that a presentable category 𝒞\mathcal{C} is a reflective localization 𝒞PSh(S)\mathcal{C} \hookrightarrow PSh(S) of a category of presheaves over SS. Since here PSh(S)PSh(S) is the free colimit completion of SS and the localization imposes relations, this is a presentation of 𝒞\mathcal{C} by generators and relations, hence the name (locally) presentable category.

See also at locally presentable categories - introduction.

Definition

There are many equivalent characterizations of locally presentable categories. The following is one of the most intuitive, equivalent characterizations are discussed below.

Definition

(locally presentable category)

A category 𝒞\mathcal{C} is called locally presentable if

  1. it is an accessible category;

  2. it has all small colimits.

This means

  1. 𝒞\mathcal{C} is a locally small category;

  2. 𝒞\mathcal{C} has all small colimits;

  3. there exists a small set SObj(𝒞)S \hookrightarrow Obj(\mathcal{C}) of objects that generates 𝒞\mathcal{C} under colimits

    (meaning that every object of 𝒞\mathcal{C} may be written as a colimit over a diagram with objects in SS);

  4. every object in 𝒞\mathcal{C} is a small object (assuming 3, this is equivalent to the assertion that every object in SS is small).

Remark

The locally in locally presentable category refers to the fact that it is the objects that are presentable, not the category as such.

For instance, consider the notion of “locally finitely presentable category”, def. 3 below, in which the generating set SS consists of finitely presentable objects, i.e. ω\omega-small ones. If one dropped the word “locally” then one would get the notion “finitely presentable category” which means something completely different, namely a finitely presentable (ω\omega-small) object of Cat.

Remark

Since a small object is one which is κ\kappa-compact for some κ\kappa, and any κ\kappa-compact object is also λ\lambda-compact for any λ>κ\lambda\gt\kappa, it follows that there exists some κ\kappa such that every object of the colimit-generating set SS is κ\kappa-compact.

This provides a “stratification” of the class of locally presentable categories, as follows.

Definition

(locally κ\kappa-presentable category)

For κ\kappa a regular cardinal, a locally κ\kappa-presentable category is a locally presentable category, def. 1, such that the colimit-generating set SS may be taken to consist of κ\kappa-compact objects.

Remark

Thus, a locally presentable category is one which is locally κ\kappa-presentable for some regular cardinal κ\kappa (hence also for every λ>κ\lambda\gt\kappa). In fact, in this case the fourth condition is redundant; once we know that there is a colimit-generating set consisting of κ\kappa-compact objects, it follows automatically that every object is λ\lambda-compact for some λ\lambda (though there is no uniform upper bound on the required size of λ\lambda). Moreover, colimit-generation is also stronger than necessary; it suffices to have a strong generator consisting of small objects.

Definition

A locally 0{}_0-presentable category is called a locally finitely presentable category.

Properties

Equivalent characterizations

There are various equivalent characterizations of locally presentable categories.

As limit-preserving functor categories

Proposition

(as limit sketches)

Locally presentable categories are precisely the categories of models of limit-sketches.

This is (Adamek-Rosicky, corollary 1.52).

Restricted to locally finitely presentable categories this becomes:

Proposition

Locally finitely presentable categories, def. 3, are equivalently the categories of finite limit preserving functors CSetC \to Set, for small finitely complete categories CC.

For the more detailed statement see below at Gabriel-Ulmer duality. Equivalently this says that:

Remark

Locally finitely presentable categories are equivalently models of finitary essentially algebraic theories.

As localizations of presheaf categories

Proposition

(as accessible reflective subcategories of presheaves)

Locally presentable categories are precisely the accessibly embedded full reflective subcategories

(Li):CiLPSh(K) (L \dashv i) : C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}} PSh(K)

of categories of presheaves on some category KK.

This appears as (Adámek-Rosický, prop 1.46).

Remark

Here accessibly embedded means that CPsh(K)C \hookrightarrow Psh(K) is an accessible functor, which in turn means that CC is closed in Psh(K)Psh(K) under κ\kappa-filtered colimits for some regular cardinal κ\kappa.

Finite presentability and Gabriel–Ulmer duality

Definition

Write LexLex for the 2-category of small categories with finite limits, with finitely continuous (i.e., finite limit preserving) functors between them, and natural transformations between those.

Write LFPLFP for the 2-category of locally finitely presentable categories, def. 3, right adjoint functors which preserve filtered colimits, and natural transformations between them.

Theorem

(Gabriel-Ulmer duality)

There is an equivalence of 2-categories

Lex opLFP Lex^{op} \stackrel{\simeq}{\to} LFP
CLex(C,Set) C \mapsto Lex(C,Set)

which sends a finitely complete category CC to the category of models of CC, i.e., the category of left exact functors CC \to Set.

Stability of presentability under various operations

Lemma

A slice category of a locally presentable category is again locally presentable.

This appears for instance as (Centazzo-Rosický-Vitale, remark 3).

Theorem

If AA is locally presentable and CC is a small category, then the functor category A CA^C is locally presentable.

Examples and applications

Locally finitely presentable categories

We list examples of locally finitely presentable categories, def. 3.

Example

The category Set of sets is locally finitely presentable.

For notice that every set is the directed colimit over the poset of all its finite subsets.

Moreover, a set SSetS \in Set is a κ\kappa-compact object precisely if it has cardinality S<κ|S| \lt \kappa. So all finite sets are 0_0-compact.

Hence a a set of generators that exhibits SetSet as a locally finitely complete category is given by the set containing one finite set of cardinality nn \in \mathbb{N} for all nn.

Example

More generally, for CC any small category the category of presheaves Set CSet^C is locally finitely presentable if CC is small.

This follows with Gabriel-Ulmer duality: the finite limit completion of CC, Lex(C)Lex(C), is also small, and Set CSet^C is equivalent to the category of finitely continuous functors Lex(C)SetLex(C) \to Set.

Example

More generally still, if AA is locally finitely presentable and CC is small, then A CA^C is locally finitely presentable.

To see this, embed AA as a finitely-accessible reflective subcategory of a presheaf topos Set BSet^B, and then note that by 2-functoriality of () C(-)^C we get A CA^C as a finitely-accessible reflective subcategory of Set B×CSet^{B \times C}.

Example

The category of algebras of a Lawvere theory, for example Grp, is locally finitely presentable. A TT-algebra AA is finitely presented if and only if the hom-functor Alg T(A,)Alg_T(A, -) preserves filtered colimits, and any TT-algebra can be expressed as a filtered colimit of finitely presented algebras.

Example

The category of coalgebras over a field kk is locally finitely presentable; similarly the category of commutative coalgebras over kk is locally finitely presentable.

Example

A poset, regarded as a category, is locally finitely presentable if it is a complete lattice which is algebraic (each element is a directed join of finite elements).

Counterexamples
  • the category FinSet of finite sets is not locally finitely presentable, as it does not have all countable colimits.

  • Top is not locally finitely presentable.

  • The opposite category of a locally presentable category (in particular, a locally finitely presentable category) is never locally presentable, unless it is a poset. This is Gabriel-Ulmer, Satz 7.13.

Locally presentable categories

Example

A poset, considered as a category, is locally presentable precisely if it is a complete lattice.

Example

The following three examples, being presheaf categories, are locally finitely presentable, thus a fortiori locally presentable. They are important for the general study of (∞,1)-categories.

More generally,

Proposition

Every sheaf topos is locally presentable.

This appears for instance as (Borceux, prop. 3.4.16, page 220). It follows directly with prop. 3 and using that every sheaf topos is an accessibly embedded subtopos of a presheaf topos

The main ingredient of a direct proof is:

Proposition

For CC a site and κ\kappa a regular cardinal strictly larger than the cardinality of Mor(C)Mor(C), every κ\kappa-filtered colimit in the sheaf topos Sh(C)Sh(C) is computed objectwise.

This implies that all representables in a sheaf topos are κ\kappa-compact objects.

Theorem

If TT is an accessible monad (a monad whose underlying functor is an accessible functor) on a locally presentable category AA, then the category A TA^T of algebras over the monad is locally presentable. In particular, if AA is locally presentable and i:BAi: B \to A is a reflective subcategory, then BB is locally presentable if ii is accessible.

This appears in (Adamek-Rosicky, 2.78).

This is actually somewhat subtle and gets into some transfinite combinatorics, from what I can gather.

Combinatorial model categories

A combinatorial model category is a model category that is in particular a locally presentable category.

Orthogonal subcategory problem

Given a class of morphisms Σ\Sigma in a locally presentable category, the answer to the orthogonal subcategory problem for Σ \Sigma^\perp is affirmative if Σ\Sigma is small, and is affirmative for any class Σ\Sigma assuming the large cardinal axiom known as Vopenka's principle.

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes\hookrightarrowalgebraic lattices\simeq Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes\hookrightarrowlocally presentable categories\simeq Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories\hookrightarrowaccessible categories
model category theorymodel toposes\hookrightarrowcombinatorial model categories\simeq Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes\hookrightarrowlocally presentable (∞,1)-categories\simeq
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories\hookrightarrowaccessible (∞,1)-categories

References

The definition is due to

  • P. Gabriel and F. Ulmer, Lokal präsentierbare Kategorien, Springer LNM 221, 1971

The standard textbook is

More details are in

Some further discussion is in

  • Francis Borceux, Handbook of Categorical Algebra: III Categories of Sheaves (proposition 3.4.16), page 220.

See also section A.1.1 of

where locally presentable categories are called just presentable categories.

Revised on January 5, 2014 08:54:37 by Urs Schreiber (89.204.154.192)