nLab
accessible (infinity,1)-category

Idea

The notion of accessible $(\mathrm{\infty }\mathrm{,1})$-category is the generalization of the notion of accessible category from category theory to (∞,1)-category theory.

It is a means to handle $(\mathrm{\infty }\mathrm{,1})$-categories that are not essentially small in terms of small data.

An accessible $(\mathrm{\infty }\mathrm{,1})$-category is one which may be large, but can entirely be accessed as an $(\mathrm{\infty }\mathrm{,1})$-category of “conglomerates of objects” in a small $(\mathrm{\infty }\mathrm{,1})$-category – precisely: that it is a category of $\kappa$-small ind-objects in some small $(\mathrm{\infty }\mathrm{,1})$-category $C$.

Definition

Definition

A quasi-category $C$ is accessible if it satisfies the following equivalent conditions

• for some regular cardinal $\kappa$ $C$ is equivalent to the quasi-category ${\mathrm{Ind}}_{\kappa }(C^0)$ of ind-objects for a small $(\mathrm{\infty }\mathrm{,1})$-category $C^0$;

• it is locally small and for some regular cardinal $\kappa$ it admits $\kappa$-filtered colimits and is generated under such from the full sub-quasi-category of $\kappa$-compact objects, which is itself an essentially small (∞,1)-category;

• for some regular cardinal $\kappa$, $C$ admits small $\kappa$-filtered colimits and contains an essentially small full subcategory which consists of $\kappa$-compact objects and generates $C$ under small $\kappa$-filtered colimits;

• it is an idempotent-complete quasi-category.

The equivalence of these conditions is discussed below.

An (∞,1)-functor between accessible $(\mathrm{\infty }\mathrm{,1})$-categories that preserves $\kappa$-filtered colimits is called an accessible (∞,1)-functor .

Properties

Equivalent characterizations

Theorem

The characterizations of accessible $(\mathrm{\infty }\mathrm{,1})$-categories are indeed all equivalent.

Proof

For the first few this is HTT, prop. 5.4.2.2. For the last one this is in section 5.4.3.

Stability under various operations

Theorem

If $C$ is an accessible quasi-category then so are

• for $K$ a small simplicial set the (∞,1)-category of (∞,1)-functors $\mathrm{Func}(K,C)$;

• for $p:K\to C$ a small diagram, the over quasi-category $C_{/p}$ and under-quasi-category $C_{p/}$.

Proof

This is HTT section 5.4.4, 5.4.5 and 5.4.6.

Theorem

The homotopy pullback of accessible quasi-categories (in the model structure for quasi-categories) is again accessible.

Proof

This is HTT, section 5.4.6.

The $(\mathrm{\infty }\mathrm{,1})$-category of accessible $(\mathrm{\infty }\mathrm{,1})$-categories

Definition

Write $\mathrm{Acc}(\mathrm{\infty }\mathrm{,1})\mathrm{Cat}\subset (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$ for the 2-sub-(∞,1)-category of (∞,1)Cat on

• those objects that are accessible $(\mathrm{\infty }\mathrm{,1})$-categories;

• those morphisms for which there is a $\kappa$ such that the (∞,1)-functor is $\kappa$-continuous and preserves $\kappa$-compact objects.

So morphisms are the accessible (∞,1)-functors that also preserves compact objects. (?)

This is HTT, def. 5.4.2.16.

Proposition

The full sub-quasi-category $\mathrm{Acc}(\mathrm{\infty }\mathrm{,1})\mathrm{Cat}\hookrightarrow (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$ is a reflective sub-(∞,1)-category

Proof

This is HTT, section 5.4.2.18.

References

The theory of accessible 1-categories is described in

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

The theory of accessible $(\mathrm{\infty }\mathrm{,1})$-categories is the topic of section 5.4 of

• Jacob Lurie, Higher Topos Theory