# nLab accessible (infinity,1)-category

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## 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 $C$.

An accessible $(\infty,1)$-category which is also locally presentable is called a compactly generated (∞,1)-category.

## Definition

Let $\kappa$ be a regular cardinal. spring

###### 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. has all $\kappa$-filtered colimits

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

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

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

1. has all $\kappa$-filtered colimits

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

4. $\mathcal{C}$ 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.

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes$\hookrightarrow$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$\hookrightarrow$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$\hookrightarrow$accessible categories
model category theorymodel toposes$\hookrightarrow$combinatorial model categories$\simeq$ Dugger’s theoremleft 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

## References

The theory of accessible 1-categories is described in

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

Revised on February 15, 2014 04:46:25 by Urs Schreiber (89.204.154.124)