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.

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

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

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exact localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes\hookrightarrowalgebraic lattices\simeq Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes\hookrightarrowlocally presentable categories\simeq Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories\hookrightarrowaccessible categories
model category theorymodel toposes\hookrightarrowcombinatorial model categories\simeq Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes\hookrightarrowlocally presentable (∞,1)-categories\simeq
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories\hookrightarrowaccessible (∞,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 February 15, 2014 04:46:25 by Urs Schreiber (