nLab
locally presentable category

Idea

A locally presentable category is one where every object is a colimit over a set of small objects.

Definition

There are many equivalent definitions of locally presentable categories. The following is one of the most intuitive.

Definition (locally presentable category)

A category $C$ is called locally presentable if

• $C$ is locally small category;

• it has all small colimits;

• there exists a small set $S$ of objects that generates $C$ under colimits (in the sense that every object of $C$ may be written as a colimit over a diagram with objects in $S$); and

• every object is a small object.

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 such that the colimit-generating set $S$ may be taken to consist of $\kappa$-compact objects.

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.

A locally $(\kappa ={\aleph }_0)$-presentable category is called a locally finitely presentable category.

Equivalent definitions

There are many other equivalent definitions of locally presentable categories.

Proposition (as limit sketches)

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

Proof

This is theorem xyz in AdRo .

This proposition extrapolates part of the Gabriel–Ulmer duality (see below), which concerns locally finitely presentable categories:

Proposition (as finite limit sketches)

Locally finitely presentable categories are precisely the categories of finite limit preserving functors $C\to \mathrm{Set}$, for small finitely complete categories $C$.

Proposition (as accessible reflective subcategories of presheaves)

Locally presentable categories are precisely the accessively 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 $K$

Where being accessibly embdeed 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 $\kappa$.

Proof

This appears as proposition 1.46 of

• Adamek, Rosicky, Locally presentable and accessible categories Cambridge University Press, (1994)

Gabriel–Ulmer duality

Let $\mathrm{Lex}$ denote the 2-category of small finitely complete categories, finitely continuous (i.e., finite limit preserving) functors, and natural transformations between them.

Let $\mathrm{LFP}$ denote the 2-category of locally finitely presentable categories, right adjoint functors which preserve filtered colimits, and natural transformations between them.

Theorem (Gabriel–Ulmer duality)

There is a contravariant biequivalence

$${\mathrm{Lex}}^{\mathrm{op}}\sim \mathrm{LFP}$$Lex^{op} \sim LFP

which sends a finitely complete category $C$ to the category of models of $C$, i.e., the category of left exact functors $C\to \mathrm{Set}$.

This deserves much further expansion, as it is a basic starting point for the more general study of locally presentable categories.

Variations

• The generalization of the concept to the context of (infinity,1)-categories is presentable (infinity,1)-category.

Terminology

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,” in which the generating set $S$ consists of finitely presentable objects, i.e. $\omega$-small ones. If you drop the word “locally” then you would get “finitely presentable category” which means something completely different, namely a finitely presentable ($\omega$-small) object of Cat.)

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

Properties

Lemma

Categories of models of finitary essentially algebraic theories are precisely equivalent to locally finitely presentable categories.

Lemma

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

This appears for instance as (CentazzoRosickyVitale, remark 3)

Examples and applications

Locally finitely presentable categories

The locally $\kappa$-presentable categories for $\kappa ={\aleph }_0$.

• Set is locally finitely presentable:

  • as the set of generators take the set containing one finite set of cardinality $n\in \mathbb{N}$ for all $n$.

  • every set is the directed colimit over the poset of all its finite subsets.

  • 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 ${\aleph }_0$-compact.

• More generally, by Gabriel–Ulmer duality, ${\mathrm{Set}}^C$ is locally finitely presentable if $C$ is small. For, 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 $\mathrm{Lex}(C)\to \mathrm{Set}$.

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

Citation: paper by Lack and Power. Will follow up on this at some point.

• 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.

This deserves to be expanded upon.

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

• 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 of a locally presentable category (in particular, a locally finitely presentable category) is never locally presentable, unless it is a poset.

Locally presentable categories

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

The following three examples, being presheaf categories, are locally finitely presentable, thus a fortiori locally presentable. They are important for the general study of $(\mathrm{\infty },1)$-categories.

• the category SSet of simplicial sets;

• the category $\mathrm{dSet}$ of dendroidal sets.

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

More generally,

Proposition

A Grothendieck topos is locally presentable.

This appears as Borceux, prop. 3.4.16, page 220.

The main ingredient in the 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 are $\kappa$-compact objects.

More generally still, we have the following stability theorems:

Theorem

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

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.

References

The definition is due to

• P. Gabriel and F. Ulmer (1971)

The standard textbook is

• Jiri Adamek, Jiri Rosicky, Locally presentable and accessible categories

More details are in

• C. Centazzo, Jiri Rosicky, 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.