accessible (infinity,1)-category



The notion of accessible (,1)(\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 (,1)(\infty,1)-categories that are not essentially small in terms of small data.

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

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


Let κ\kappa be a regular cardinal. spring


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

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

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

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

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

    1. is locally small

    2. has all κ\kappa-filtered colimits

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

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

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

    1. is locally small

    2. has all κ\kappa-filtered colimits

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

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

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

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


These conditions are indeed equivalent.

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


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


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

  • those objects that are accessible (,1)(\infty,1)-categories;

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

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

This is HTT, def.


Stability under various operations


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

This is HTT section 5.4.4, 5.4.5 and 5.4.6.


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

This is HTT, section 5.4.6.



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

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

preserves these.

This is HTT, proposition

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

(n,r)-categoriestoposeslocally 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)-topos theory(∞,1)-toposeslocally presentable (∞,1)-categoriesSimpson’s theorem(∞,1)-presheaf (∞,1)-categoriesaccessible (∞,1)-categories


The theory of accessible 1-categories is described in

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

Revised on March 20, 2016 10:57:36 by Tim Campion? (