# nLab accessible (infinity,1)-category

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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 $C$.

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. 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 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.

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 presentableloc finitely preslocalization theoremfree cocompletionaccessible
(0,1)-category theorylocalessuplatticealgebraic latticesPorst’s theorempowersetposet
category theorytoposeslocally presentable categorieslocally finitely presentable categoriesAdámek-Rosický‘s theorempresheaf categoryaccessible categories
model category theorymodel toposescombinatorial model categoriesDugger's theoremglobal model structures on simplicial presheavesn/a
(∞,1)-category theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories

Theory of accessible 1-categories:

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