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.

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.

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.

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.

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: 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

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 March 20, 2016 10:57:36 by Tim Campion? (108.219.95.109)