nLab
accessible (infinity,1)-category

Contents

Idea

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.

Definition

Let κ\kappa be a regular cardinal. spring

Definition

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.

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

Definition

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. 5.4.2.16.

Properties

Stability under various operations

Theorem

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.

Theorem

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

This is HTT, section 5.4.6.

Generally:

Theorem

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 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\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

References

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 (89.204.154.124)