nLab
locally presentable category

Idea

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

This says equivalently that a presentable category $𝒞$ is a reflective localization $𝒞\hookrightarrow \mathrm{PSh}(S)$ of a category of presheaves over $S$. Since here $\mathrm{PSh}(S)$ is the free colimit completion of $S$ and the localization imposes relations, this is a presentation of $𝒞$ 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 $𝒞$ is called locally presentable if

1. it is an accessible category;

2. it has all small colimits.

This means

1. $𝒞$ is a locally small category;

2. $𝒞$ has all small colimits;

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

   (meaning that every object of $𝒞$ may be written as a colimit over a diagram with objects in $S$);

4. every object in $𝒞$ is a small object (assuming 3, this is equivalent to the assertion that every object in $S$ 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 $S$ consists of finitely presentable objects, i.e. $\omega$-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 ($\omega$-small) object of Cat.

Remark

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

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

Definition

(locally $\kappa$-presentable category)

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

Remark

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

Definition

A locally ℵ${}_0$-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 $C\to \mathrm{Set}$, for small finitely complete categories $C$.

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

$$(L⊣i):C\stackrel{\stackrel{L}{\leftarrow }}{\underset{i}{\hookrightarrow }}\mathrm{PSh}(K)$$(L \dashv i) : C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}} PSh(K)

of categories of presheaves on some category $K$.

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

Remark

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

Finite presentability and Gabriel–Ulmer duality

Definition

Write $\mathrm{Lex}$ 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 $\mathrm{LFP}$ 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

$${\mathrm{Lex}}^{\mathrm{op}}\stackrel{\simeq }{\to }\mathrm{LFP}$$Lex^{op} \stackrel{\simeq}{\to} LFP

$$C\mapsto \mathrm{Lex}(C,\mathrm{Set})$$C \mapsto Lex(C,Set)

which sends a finitely complete category $C$ to the category of models of $C$, i.e., the category of left exact functors $C\to$ 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 $A$ is locally presentable and $C$ is a small category, then the functor category $A^C$ 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 $S\in \mathrm{Set}$ is a $\kappa$-compact object precisely if it has cardinality $\mid S\mid <\kappa$. So all finite sets are ℵ${}_0$-compact.

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

Example

More generally, for $C$ any small category the category of presheaves ${\mathrm{Set}}^C$ is locally finitely presentable if $C$ is small.

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

Example

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

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

Example

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

Example

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

• the category sSet of simplicial sets;

• the category dSet of dendroidal sets.

• for $C$ a small category the functor category $\mathrm{Funct}(C,\mathrm{sSet})$ of simplicial presheaves.

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

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

Theorem

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

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

Related concepts

• Another notion of “presentable category” is that of an equationally presentable category.

• Locally presentable categories are a special case of locally bounded categories.

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

(n,r)-categories…	satisfying Giraud's axioms	inclusion of left exaxt localizations	generated under colimits from small objects		localization of free cocompletion		generated under filtered colimits from small objects
(0,1)-category theory	(0,1)-toposes	$\hookrightarrow$	algebraic lattices	$\simeq$ Porst’s theorem	subobject lattices in accessible reflective subcategories of presheaf categories
category theory	toposes	$\hookrightarrow$	locally presentable categories	$\simeq$ Adámek-Rosický’s theorem	accessible reflective subcategories of presheaf categories	$\hookrightarrow$	accessible categories
model category theory	model toposes	$\hookrightarrow$	combinatorial model categories	$\simeq$ Dugger’s theorem	left Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory	(∞,1)-toposes	$\hookrightarrow$	locally presentable (∞,1)-categories	$\simeq$ Simpson’s theorem	accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories	$\hookrightarrow$	accessible (∞,1)-categories

References

The definition is due to

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

The standard textbook is

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

More details are in

• C. Centazzo, Jiří Rosický, Enrico Vitale, A characterization of locally $𝔻$-presentable categories (pdf)

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

• Jacob Lurie, Higher Topos Theory

where locally presentable categories are called just presentable categories.