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 nice colimit over objects in this set.

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

1. it is an accessible category;

2. it has all small colimits.

This means

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

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

3. there exists a small set $S \hookrightarrow Obj(\mathcal{C})$ of $\lambda$-compact objects that generates $\mathcal{C}$ under $\lambda$-filtered colimits for some regular cardinal $\lambda$.

Remark

If follows that every object in a locally presentable category is a small object.

Remark

The locally in locally presentable category refers to the fact that it is the objects of (in) the category that are presentable, not the category as such (which is itself an object of Cat). This permits the distinction between, for instance, locally presentable categories, and finitely presentable objects of Cat, which could be called “finitely presentable categories”. In practice, however, it is common to drop “locally” from “locally presentable category” without modifying the meaning.

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 $\lambda \gt \kappa$, 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. , 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 $\lambda\gt\kappa$). 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 ${\aleph}_0$-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 Adámek & Rosický (1994), corollary 1.52.

Restricted to locally finitely presentable categories this becomes:

Proposition

Locally finitely presentable categories, def. , are equivalently the categories of finite limit preserving functors $C \to Set$, 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

$$(L \dashv i) \;\colon\; C \underoverset {\underset{i}{\hookrightarrow}} {\overset{L}{\leftarrow}} {\;\;\bot\;\;} PSh(K)$$

of categories of presheaves on some small category $K$.

This appears as Adámek & Rosický (1994), prop 1.46 and Gabriel & Ulmer (1971), Satz 8.5(c) and Bemerkung 8.6(c).

Remark

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

See also at sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes.

Corollary

Locally presentable categories are complete.

Proof

A reflective subcategory of a complete category is complete, since monadic functors reflect limits, and the above proposition shows that any locally presentable category is a reflective subcategory of a presheaf category, which is complete.

Finite presentability and Gabriel–Ulmer duality

Definition

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

Theorem

(Gabriel-Ulmer duality)

There is an equivalence of 2-categories

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

$$C \mapsto Lex(C,Set)$$

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.

Well-poweredness and well-copoweredness

• Every locally presentable category is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.

• Every locally presentable category is also well-copowered. This is shown in Adámek & Rosický (1994), Prop. 1.57 & Thm. 2.49.

Examples and applications

Locally finitely presentable categories

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

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 $S \in Set$ is a $\kappa$-compact object precisely if it has cardinality $|S| \lt \kappa$. So all finite sets are $\aleph_0$-compact.

Hence a a set of generators that exhibits $Set$ 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 $Set^C$ is locally finitely presentable.

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

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

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 $Alg_T(A, -)$ 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).

Remark

(counter-examples)

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

• The category Field of fields and field homomorphisms is not locally presentable, as it does not have all binary coproducts (for instance, there are none between fields of differing characteristics).

• TopologicalSpaces 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

Every Grothendieck abelian category is locally presentable [Beke (200), Prop. 3.10, cf. Krause (2015), Cor. 5.2].

This implies in particular (by this example at Grothendieck abelian category) that for $R$ a commutative ring (internal to any Grothendieck topos):

• the category $R$Mod of $R$-modules is locally presentable

• the category $Ch_\bullet(R Mod)$ of chain complexes is locally presentable.

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 $Funct(C,sSet)$ of simplicial presheaves

  (see Prop. below).

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. and using that every sheaf topos is an accessibly embedded subtopos of a presheaf topos (see at sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes)

The main ingredient of a direct proof is:

Proposition

For $C$ a site and $\kappa$ a regular cardinal strictly larger than the cardinality of $Mor(C)$, every $\kappa$-filtered colimit in the sheaf topos $Sh(C)$ 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 appears in Adámek & Rosický (1994), 2.78.

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

Proposition

Given

• $\mathcal{C}$ a small category,

• $\mathcal{A}$ a locally presentable category

then also the functor category $Func(\mathcal{C}, \mathcal{A})$ is locally presentable.

This is Adámek & Rosický (1994), Cor. 1.54

See at Functor category – Local presentability for more.

Proposition

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

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

Proposition

(locally presentable Grothendieck constructions)
Given a pseudofunctor with values in $Cat_{Adj}$ as

$$\array{ \mathllap{ \mathbf{C} \,\colon\, \; } Base &\longrightarrow& Cat \\ \mathcal{X} &\mapsto& \mathbf{C}_{\mathcal{X}} \\ \Big\downarrow\mathrlap{{}^{f}} && \mathllap{^{f^\ast}}\Big\uparrow \Big\downarrow\mathrlap{{}^{f_!}} \\ \mathcal{Y} &\mapsto& \mathbf{C}_{\mathcal{Y}} }$$

such that for some regular cardinal $\kappa$

1. $Base$ is locally $\kappa$-presentable,

2. each $\mathbf{C}_{\mathcal{X}}$ is locally presentable,

3. $\mathbf{C}_{(-)}$ preserves $\kappa$-filtered 2-limits

then also the Grothendieck construction $\int \mathbf{C}$ is locally presentable.

This follows, as explained in MO:a/102083, from the analogous statement for accessibility which appears as Makkai & Paré (1989), Prop. 5.3.1. (4).

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

• ind-object

• PrCat, Pr(∞,1)Cat, Ho(CombModCat)

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

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

• class-locally presentable category

• locally strongly finitely presentable category

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

General

The definition is due to

• Pierre Gabriel, Friedrich Ulmer, Lokal präsentierbare Kategorien, Springer LNM 221 (1971) [doi:10.1007/BFb0059396, MR0327863]

Textbook account:

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189, Cambridge University Press (1994) [doi:10.1017/CBO9780511600579]

Review for the case of locally finitely presentable categories:

• Maru Sarazola, An introduction to locally finitely presentable categories (2017) [pdf, pdf]

See also:

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

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

• Jacob Lurie, A.1.1 of: Higher Topos Theory

  (where locally presentable categories are called presentable categories)

On the example of Grothendieck abelian categories:

• Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philosophical Society 129 3 (2000) 447-475 [arXiv:math/0102087, doi:10.1017/S0305004100004722]

• Greg Bird, Limits in 2-Categories of Locally-Presentable Categories pdf]

• Henning Krause, Deriving Auslander’s formula, Documenta Math. 20 (2015) 669-688 [arXiv:1409.7051]

In enriched category theory

Discussion of local presentability in enriched category theory (see also references on enriched accessible categories):

• G. Max Kelly, Structures defined by finite limits in the enriched context, I, Cahiers de topologie et géométrie différentielle 23 1 (1982) 3-42 [numdam:CTGDC_1982__23_1_3_0]

See also:

• MO:q/53470: Enriched locally presentable categories