on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
equivalences in/of $(\infty,1)$-categories
This page means to give an introduction to the notion of locally presentable category, and its related notions in higher category theory and survey some fundamental properties.
Expected background of the reader:
For the first section Basic idea in category theory the reader is assumed to be familiar with basic notions of category theory such as presheaves and colimits.
For the section Basic idea in model category theory the reader is assumed to be familiar with basic notions in model category theory such as cofibrantly generated model categories and homotopy colimits.
For the third section Basic idea in (∞,1)-category theory the reader is assumed to be familiar with basic concepts of (∞.1)-category theory such as (∞,1)-categories of (∞,1)-presheaves and (∞,1)-colimits.
The general idea is that a locally presentable category is a large category generated from small data: from small objects under small colimit.
The notion of locally presentable category is, at least roughly, an analogue for categories of the notion of a finitely generated module.
An abelian group $A$ is called finitely generated if there is a finite subset
of the underlying set $U(A)$ of $A$, such that every element of $A$ is a sum of such generating elements.
We always have the maximal such presentation where $S = U(A)$ is the whole underlying set and $\iota \colon F(U(A)) \to A$ is the counit of the free-forgetful adjunction. But the presentation is all the more interesting/useful the smaller $S$ is.
Now, the categorification of “commutative sum” is colimit. Hence let now $\mathcal{C}$ be a category with all small colimits.
We say a subclass $S \hookrightarrow Obj(\mathcal{C})$ of objects or equivalently the full subcategory $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on this subclass generates $\mathcal{C}$ if every object in $\mathcal{C}$ is a colimit of objects in $\mathcal{C}^0$, hence the colimit over a diagram of the form
As before, such a presentation is all the more useful the “smaller” the generating data is. In order to grasp the various aspects of the notion of “smallness” in category theory we need to recall the notion of regular cardinal.
The cardinality $\kappa = {\vert S\vert}$ of a set $S$ is regular if every coproduct/disjoint union of sets of cardinality smaller than $\kappa$ and indexed by a set of cardinality smaller than $\kappa$ is itself of cardinality smaller than $\kappa$.
The smallest regular cardinal is ℵ${}_0 = {\vert \mathbb{N}\vert}$: every finite union of finite sets is itself a finite set. (See the entry on regular cardinals for a discussion as to whether one might consider some finite cardinals as being `regular'.)
We can now speak of objects that are “$\kappa$-small sums” using the notion of $\kappa$-filtered colimits:
For $\kappa$ a regular cardinal, a $\kappa$-filtered category is one where every diagram of size $\lt \kappa$ has a cocone.
In an ℵ${}_0$-filtered category every finite diagram has a cocone. This is equivalent to:
for every pair of objects there is a third objct such that both have a morphism to it;
for every pair of parallel morphisms there is a morphism out of their codomain such that the two composites are equal.
The tower diagram category $(\mathbb{N}, \leq)$
is filtered.
For $\lambda \gt \kappa$ a bigger regular cardinal, every $\lambda$-filtered category is in particular also $\kappa$-filtered.
Using this we have the central definition now:
A $\kappa$-filtered colimit is a colimit over a $\kappa$-filtered diagram.
A crucial characterizing property of $\kappa$-filtered colimits is the following:
A colimit in Set is $\kappa$-filtered precisely if it commutes with all $\kappa$-small limits.
In particular a colimit in Set is filtered (meaning: ℵ${}_0$-filtered) precisely if it commutes with all finite limits.
An object $A \in \mathcal{C}$ is a $\kappa$-compact object if it commutes with $\kappa$-filtered colimits, hence if for $X \colon I \to \mathcal{C}$ any $\kappa$-filtered diagram, the canonical function
is a bijection.
We say $X$ is a small object if it is $\kappa$-compact for some regular cardinal $\kappa$.
If $\lambda \gt \kappa$, then being $\lambda$-compact is a weaker condition that being $\kappa$-comopact.
The object $A$ commutes with the colimit over $I$ precisely if every morphism $A \to \underset{\to_i}{\lim} X_i$ lifts to a morphism $A \to X_j$ into one of the $X_j$. Schematically, depicting specifically a sequential colimit, this means that we have:
Hence $A$ is “small enough” such that mapping it into the sum of all the $X_i$ it always entirely lands inside one of the $X_i$.
There is a close relation between the notion of “compact” as in, on the one hand, compact topological space and compact topos, and on the other as in compact object as above. This is mediated by proposition 1. But the relation is a bit more subtle and takes a bit more discussion than we maybe want to get into here.
Using this we can now say:
A locally small category $\mathcal{C}$ is a locally presentable category if it has all small colimits and there is a small set $S \hookrightarrow Obj(\mathcal{C})$ of small objects such that this generates $\mathcal{C}$, by def. 1.
The adjective “locally” in “locally presentable category” is to indicate that the condition is all about the objects, only. There is a different notion of “presented category”.
There are a bunch of equivalent reformulations of the notion of locally presentable category. One of the more important ones we again motivate first by analogy with presentable modules:
If an abelian group $A$ is generated by a set $S \hookrightarrow U(A)$ as in example 1, this means equivalently that there is an epimorphism
from the free abelian group $F(S)$ generated by $S$, hence the group obtained by forming formal sums of elements in $S$. Here the epimorphism sends formal sums to actual sums in $A$:
The categorification of the notion free abelian group is the notion of free cocompletion of a category $\mathcal{C}^0$: the category of presheaves $PSh(\mathcal{C}^0)$.
Accordingly:
If a full subcategory $\iota \colon \mathcal{C}^0 \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under colimits as in defn. 1, then there is a functor
which sends formal colimits to actual colimits in $\mathcal{C}$
Here $L$ by construction preserves all colimits.
Therefore conversely, given a colimit-preserving functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ we want to say that it locally presents $\mathcal{C}$ if $L$ is “suitably epi”.
It turns out that “suitably epi” is to be the following:
A functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from the category of presheaves over a small category $\mathcal{C}^0$ is an accessible localization if
$L$ has a section, hence a functor $R \colon \mathcal{C} \to PSh(\mathcal{C}^0)$ with a natural isomorphism $L\circ R \simeq id_{\mathcal{C}}$;
such that
$R$ is right adjoint to $L$;
$R\circ L$ preserves $\kappa$-filtered colimits.
With this notion we have the following analog of the familiar statement that an abelian group is generated by $S$ precisely if there is an epimorphism $L \colon F(S) \to A$:
A category $\mathcal{C}$ is locally presentable according to def. 6 precisely if it is an accessible localization, def. 7,
for some small category $\mathcal{C}^0$.
This is due to (Adámek-Rosický, prop 1.46).
A locally presentable category $\mathcal{C}$ is called a topos, precisely if the localization functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from theorem 1 in addition is a left exact functor, meaning that it preserves finite limits.
In summary the discussion above says that the notion of locally presentable categories sits in a sequence of notions as indicated in the row labeled “category theory” in the following table. The other rows are supposed to indicate that regarding a category as a (1,1)-category and simply varying in this story the parameters $(n,r)$ in “(n,r)-category” one obtains fairly straightforward analogs of the notion of locally presentable category in other fragments of higher category theory. These we discuss in more detail further below.
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 |
The analog of a category of presheaves in model category theory is the model structure on simplicial presheaves, which we now briefly indicate.
Write sSet for the category of simplicial sets. Here we always regard this as equipped with the standard model structure on simplicial sets $sSet_{Quillen}$.
For $C$ a small category write $[C^{op}, sSet]\simeq [C^{op}, Set]^{\Delta^{op}}$ for the category of simplicial presheaves. The global projective model structure on simplicial presheaves $[C^{op}, sSet]_{proj}$ has as
weak equivalences the objectwise weak homotopy equivalences of simplicial sets
fibrations the objectwise Kan fibrations.
Accordingly $[C^{op}, sSet]$ is a cofibrantly generated model category with generating (acyclic) cofibrations the tensoring of objects of $C$ with the generating (acyclic) cofibrations of $sSet_{Quillen}$.
Given a model category $[C^{op}, Set]_{proj}$ and set $\mathcal{S} \subset Mor([C^{op}, Set])$ of morphisms, the left Bousfield localization is the model structure with the same cofibrations and weak equivalences the $\mathcal{S}$-local morphisms.
The simple idea of the following definition is to say that the model category analog of locally presentable category is simply a model structure on a locally presentable category.
A model category is a combinatorial model category if
the underlying category is a locally presentable category;
the model structure is a cofibrantly generated model category.
Every combinatorial model category, def. 10, is Quillen equivalent to a left Bousfield localization of a global model structure on simplicial presheaves as in def. 9.
See at combinatorial model category - Dugger’s theorem.
For $\mathcal{C}$ and $\mathcal{D}$ two (∞,1)-categories and $\mathcal{C}_{s}, \mathcla{D}_s \in sSet$ two models as quasi-categories, an (∞,1)-functor $F \colon \mathcal{C} \to \mathcal{D}$ is simply a homomorphism of simplicial set $\mathcal{C}_s \to \mathcal{D}_s$.
The (∞,1)-category of (∞,1)-functors $Func(\mathcal{C}, \mathcal{D})_s$ as a quasi-category is simply the hom object of simplicial set
For $\mathcal{D}$ an (∞,1)-category, the (∞,1)-category of (∞,1)-presheaves on $\mathcal{D}$ is the functor category
out of the opposite (∞,1)-category of $\mathcal{D}$ into the (∞,1)-category of ∞-groupoids.
The notions of adjoint functors, full and faithful functors etc. have straightforward, essentially verbatim generalizations to $(\infty,1)$-categories:
A pair of (∞,1)-functors
is a pair of adjoint (∞,1)-functors, if there exists a unit transformation $\epsilon : Id_C \to R \circ L$ – a morphism in the (∞,1)-category of (∞,1)-functors $Func(C,D)$ – such that for all $c \in C$ and $d \in D$ the induced morphism
is an equivalence of ∞-groupoids.
An (∞,1)-functor $F \colon \mathcal{C} \to \mathcal{D}$ is a full and faithful (∞,1)-functor if for all objects $X,Y \in \mathcal{C}$ the component
is an equivalence of ∞-groupoids.
A reflective sub-(∞,1)-category $\mathcal{C} \hookrightarrow \mathcal{D}$ is a full and faithful (∞,1)-functor with a left adjoint (∞,1)-functor.
We have then the essentially verbatim analog of the situation for ordinary categories:
An (∞,1)-category $\mathcal{C}$ is a locally presentable (∞,1)-category if there exists a small set of objects such that the full sub-(∞,1)-category $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on it generates $\mathcal{C}$ under (∞,1)-colimits.
And the equivalent characterization is now as before
An (∞,1)-category is a locally presentable (∞,1)-category, def. 16, precisely if it is equivalent to localization, def. 15,
of an (∞,1)-category of (∞,1)-presheaves, def. 12, such that $R \circ L$ preserves $\kappa$-filtered (∞,1)-colimits for some regular cardinal $\kappa$.
This appears as Lurie, theorem 5.5.1.1, attributed there to Carlos Simpson.
As before, if a locally presentable $(\infty,1)$-category arises as the localization $L \colon PSh_\infty(\mathcal{C}^0) \to \mathcal{C}$ of a left exact (∞,1)-functor, then it is an (∞,1)-topos.
There is a close match between the theory of combinatorial model categories and locally presentable (∞,1)-categories.
Every locally presentable (∞,1)-category arises, up to equivalence of (∞,1)-categories, as the simplicial localization of a combinatorial model category.
This is part of Lurie, theorem 5.5.1.1.
Accordingly, every simplicial Quillen adjunction between combinatorial model categories gives rise to a pair of adjoint (∞,1)-functors between the corresponding locally presentable $(\infty,1)$-categories.
Hence a left Bousfield localization of a model structure on simplicial presheaves presents a corresponding localization of an (∞,1)-category of (∞,1)-presheaves to a locally presentable (∞,1)-category.
The standard textbook for locally presentable categories is
Decent accounts of combinatorial model categories include secton A.2.6 of
and
The standard text for locally presentable (∞,1)-categories is section 5 of Lurie.