nLab accessible (infinity,1)-category

Idea

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

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

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

A $\kappa$-accessible $(\infty,1)$-category which in addition has all (∞,1)-colimits is called a locally ∞-presentable or a $\kappa$-compactly generated (∞,1)-category.

Definition

Let $\kappa$ be a regular cardinal.

Definition

A (∞,1)-category $\mathcal{C}$ is $\kappa$-accessible if it satisfies the following equivalent conditions:

1. There is a small (∞,1)-category $\mathcal{C}^0$ and an equivalence of (∞,1)-categories

   $$\mathcal{C} \simeq Ind_\kappa(C^0)$$

   of $\mathcal{C}$ with the (∞,1)-category of ind-objects, relative $\kappa$, in $\mathcal{C}^0$.

2. The $(\infty,1)$-category $\mathcal{C}$

   1. is locally small

   2. has all $\kappa$-filtered colimits

   3. the full sub-(∞,1)-category $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ of $\kappa$-compact objects is an essentially small (∞,1)-category;

   4. $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under $\kappa$-filtered (∞,1)-colimits.

3. The $(\infty,1)$-category $\mathcal{C}$

   1. is locally small

   2. has all $\kappa$-filtered colimits

   3. there is some essentially small$\,$ sub-(∞,1)-category $\mathcal{C}' \hookrightarrow \mathcal{C}$ of $\kappa$-compact objects which generates $\mathcal{C}$ under $\kappa$-filtered (∞,1)-colimits.

The notion of accessibility is mostly interesting for large (∞,1)-categories. For

• If $\mathcal{C}$ is small, then there exists a $\kappa$ such that $\mathcal{C}$ is $\kappa$-accessible if and only if $\mathcal{C}$ is an idempotent-complete (∞,1)-category.

Generally, $\mathcal{C}$ is called an accessible $(\infty,1)$-category if it is $\kappa$-accessible for some regular cardinal $\kappa$.

Proposition

These conditions are indeed equivalent.

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

Definition

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

Definition

Write $(\infty,1)AccCat \subset (\infty,1)Cat$ for the 2-sub-(∞,1)-category of (∞,1)Cat on

• those objects that are accessible $(\infty,1)$-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.

Properties

Stability under various operations

Theorem

If $C$ is an accessible $(\infty,1)$-category then so are

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

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

This is HTT section 5.4.4, 5.4.5 and 5.4.6.

Theorem

The (∞,1)-pullback of accessible $(\infty,1)$-categories in (∞,1)Cat is again accessible.

This is HTT, section 5.4.6.

Generally:

Theorem

The $(\infty,1)$-category $(\infty,1)AccCat$ has all small (∞,1)-limits and the inclusion

$$(\infty,1)AccCAT \hookrightarrow (\infty,1)CAT$$

preserves these.

This is HTT, proposition 5.4.7.3.

Related concepts

• compactly generated (∞,1)-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

Theory of accessible 1-categories:

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

Theory of accessible $(\infty,1)$-categories:

• Jacob Lurie, Section 5.4 of Higher Topos Theory

See also:

• Charles Rezk, Generalizing accessible ∞-categories, 2021 (pdf).