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 set $S$ (not a proper class) 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)

Every locally finitely presentable category $A$ is equivalent to the category of finite limit preserving functors $C\to \mathrm{Set}$, for some small finitely complete category $C$.

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 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 equationally presentable category.

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,

• A Grothendieck topos is locally presentable.

A proof may be found in Borceux’s Handbook of Categorical Algebra: Categories of Sheaves (proposition 3.4.16), page 220.

More generally still, we have the following stability theorems:

• If $A$ is locally presentable and $C$ is small, then $A^C$ is locally presentable.

• If $T$ is an accessible monad (a monad whose underlying functor is an accessible functor) on a locally presentable category $A$, then the category of algebras $A^T$ 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.

References

The definition is due to

• P. Gabriel and F. Ulmer (1971)

The standard textbook is

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

See also section A.1.1 of

• Jacob Lurie, Higher Topos Theory

where locally presentable categories are called just presentable categories.

Discussion

A previous version of this entry led to the following discussion.

Do you mean locally presentable? Or are you following Lurie now? Should we change the terminology below?

Urs Schreiber: oh, sorry, this is a result of bad copy-and-pasting, I had meant to type locally presentable – but now that we are discussing this, maybe we can sort this out: what’s the point of saying “locally” presentable? Which reasons would there be not to follow Lurie on this?

Toby: The only thing that I can think of offhand is to distinguish from equationally presentable category. (Actually, what I really mean is that if I Google "presentable category", then I mostly get hits with ‘locally’ in them, and if I Google "presentable category" -locally, then I mostly get hits with ‘equationally’ in them.)

Urs Schreiber: I have changed the wording in the first sentence now.

I guess the point remains that “locally presentable category” serves to distinguish from “equationally presentable category”.

On the other hand, locally presentable then still seems like a bad choice of terminology, as it indicates nothing about the kind of presentation and in fact it remains a mystery to me what is supposed to be local about the above notion. (It’s not the local smallness, or is it?)

But then it gets a bit worse even when we look at the generalizations. I have firmly followed Lurie with the terminology at presentable (infinity,1)-category that is supposed to generalize the notion here, and there it turns out to be pretty good terminology, as those $(\mathrm{\infty }\mathrm{,1})$-categories are, among a whole list of equivalent characterizations, precisely those that are given by combinatorial simplicial model categories. I have used that nice fact to consistently say “a model category presents an $(\mathrm{\infty }\mathrm{,1})$-category” with the “presents” linking to presentable $(\mathrm{\infty }\mathrm{,1})$-category .

Mike Shulman: Coming into this discussion late, let me speak up in favor of “locally presentable.” I agree that “locally” is a poor choice (and an overworked word), but I think that merely “presentable” is worse; some adjective must be used and “locally” has the weight of history behind it. The important point requiring the adjective is that it is the objects of the category which are presentable, rather than the category itself. 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 an $\omega$-small object of Cat.

I’m guessing that when you say “presentable category” you are thinking of the set $S$ as “presenting” the category in some way, but I think this is wrong. The notion of “presentable object” has a precise meaning in the very theory we are discussing, so we shouldn’t apply it unqualified to the category as well with a different meaning.

Reid Barton: I don’t find these arguments very convincing. First, local presentability is not something that can be checked objectwise, even once the category is assumed to be cocomplete. For example, take the large poset of all sets (or elements of a fixed universe), ordered by inclusion. This category is cocomplete, and every object is $\kappa$-presentable where $\kappa$ is its cardinality as a set, but it is not locally presentable: it has no terminal object.

Secondly, and less pathologically, the name “locally finitely presentable” makes no sense at all from this point of view. The category of groups is locally finitely presentable even though most of its objects are not finitely presentable. It is only the category as a whole which is finitely presentable in the sense of being specified by a finite amount of data (the localization of presheaves on a finite category at finitely many morphisms).

Finally, locally presentable categories should be viewed as a full sub-2-category not of Cat but of cocomplete categories and colimit-preserving functors. A presentation of a category as the localization of a presheaf category at a set of maps is exactly the data you need to compute Hom out of it in this 2-category. I have not worked out the details, but it seems that local presentability should be closely related to presentability in this setting.

Mike Shulman: When I said “the objects are presentable” I didn’t mean that as a definition, just as a fact which is true about locally presentable categories and may help to get intuition for the choice of terminology. I agree that “locally” is not the best choice of word, but I think some word has to be used, because saying “finitely presentable” to mean “locally finitely presentable” is even worse. A finitely presentable category means a finitely presentable object of Cat, i.e. a category which is a quotient of a finite directed graph by finitely many relations. That is an important notion, and “finitely presentable category” is the completely natural term for it, in line with all sorts of universal algebra and even with the theory of locally presentable categories.

If the term under discussion was “presentable cocomplete categories,” then I would find your arguments more convincing. A presentable category should mean a presentable object of $\mathrm{Cat}$; I think if you want to talk about presentable objects of $\mathrm{CocpltCat}$, then the terminology should indicate it. Not all categories are cocomplete.

But also, I don’t think a locally finitely presentable category is necessarily a presentable object of $\mathrm{CocpltCat}$ in the categorical sense. Let $D$ be any small directed set, for any $x\in D$ let $D/x=\{y\in D\mid y\le x\}$, and consider the $D$-indexed diagram $x\mapsto [(D/x{)}^{op},\mathrm{Set}]$ in $\mathrm{CocpltCat}$, with transitions induced by the inclusions. Now its colimit, $E$ say, contains a canonical diagram of shape $D$, where $x$ maps to the image of $x\in D/x\hookrightarrow [(D/x{)}^{op},\mathrm{Set}]\to E$, and since $E$ is cocomplete this diagram has a colimit $y$. But $y$ is not in any of the categories $[(D/x{)}^{op},\mathrm{Set}]$, and therefore (since $\mathrm{Set}$ is the free cocomplete category on one generator) there is a cocontinuous functor $\mathrm{Set}\to [(D/x{)}^{op},\mathrm{Set}]$ not factoring through any $[(D/x{)}^{op},\mathrm{Set}]$. So $\mathrm{Set}$ is not a presentable object of $\mathrm{CocpltCat}$.