nLab
accessible category

Contents

Contents

Idea

An accessible category is a possibly large category which is however essentially determined by a small category, in a certain way.

Definition

Definition

A locally small category CC is κ\kappa-accessible for a regular cardinal κ\kappa if:

  1. the category has κ\kappa-directed colimits (or, equivalently, κ\kappa-filtered colimits), and

  2. there is a set of κ\kappa-compact objects that generate the category under κ\kappa-directed colimits.

Then CC is an accessible category if there exists a κ\kappa such that it is κ\kappa-accessible.

Remark

Unlike for locally presentable categories, it does not follow that if CC is κ\kappa-accessible and κ<λ\kappa\lt \lambda then CC is also λ\lambda-accessible. It is true, however, that for any accessible category, there are arbitrarily large cardinals λ\lambda such that CC is λ\lambda-accessible.

Proposition

Equivalent characterizations include that CC is accessible iff:

  • it is the category of models (in Set) of some small sketch.

  • it is of the form Ind κ(S)Ind_\kappa(S) for SS small, i.e. the κ\kappa-ind-completion of a small category, for some κ\kappa.

  • it is of the form κFlat(S)\kappa\,Flat(S) for SS small and some κ\kappa, i.e. the category of κ\kappa-flat functors from some small category to SetSet.

  • it is the category of models (in SetSet) of a suitable type of logical theory.

The relevant notion of functor between accessible categories is

Definition

A functor F:CDF\colon C\to D between accessible categories is an accessible functor if there exists a κ\kappa such that CC and DD are both κ\kappa-accessible and FF preserves κ\kappa-filtered colimits.

Properties

Raising the index of accessibility

If CC is λ\lambda-accessible and λμ\lambda\unlhd\mu (see sharply smaller cardinal), then CC is μ\mu-accessible. Thus, any accessible category is μ\mu-accessible for arbitrarily large cardinals μ\mu.

Stability under various constructions

Proposition

If 𝒞\mathcal{C} is an accessible category and KK is a small category, then the category of presheaves Func(K op,𝒞)Func(K^{op}, \mathcal{C}) is again accessible.

(Lurie, prop. 5.4.4.3)

Proposition

(preservation of accessibility under inverse images)

Let F:CDF : C \to D be a functor between locally presentable categories which preserves κ\kappa-filtered colimits, and let D 0DD_0 \subset D be an accessible subcategory. Then the inverse image f 1(D 0)Cf^{-1}(D_0) \subset C is a κ\kappa-accessible subcategory.

This appears as HTT, corollary A.2.6.5.

Proposition

(accessibility of fibrations and weak equivalences in a combinatorial model category)

Let CC be a combinatorial model category, Arr(C)Arr(C) its arrow category, WArr(C)W \subset Arr(C) the full subcategory on the weak equivalences and FArr(C)F \subset Arr(C) the full subcategory on the fibrations. Then FF, WW and FWF \cap W are accessible subcategories of Arr(C)Arr(C).

This appears as HTT, corollary A.2.6.6.

Proposition

(closure under limits)

The 2-category AccAcc of accessible categories, accessible functors, and natural transformations has all small 2-limits.

This can be found in Makkai-Paré. Some special cases are proven in Adámek-Rosický.

Proposition

(directed unions)

The 2-category AccAcc has directed colimits of systems of fully faithful functors. If there is a proper class of strongly compact cardinals?, then it has directed colimits of systems of faithful functors.

See (Paré-Rosický).

Adjoint functor theorem

Proposition

(adjoint functors)

Every accessible functor satisfies the solution set condition, and every left or right adjoint between accessible categories is accessible. Therefore, the adjoint functor theorem takes an especially pleasing form for accessible categories: a functor is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits).

This is the familiar statement for locally presentable categories but as far as I know it isn’t true for accessible categories (because sufficient limits/colimits are needed in the ambient categories to construct the adjoint to a given functor).

Idempotence completeness

Proposition

A small category is accessible precisely when it is idempotent complete.

Makkai-Paré say that this means accessibility is an “almost pure smallness condition.”

Categories of models over a theory

Proposition

A geometric theory TT is a theory of presheaf type precisely if its category Mod(T,Set)Mod(T,Set) of models in Set is a finitely accessible category, and if and only if it is sketchable.

See also at categorical model theory.

Well-poweredness and well-copoweredness

Examples

Functor categories

See at Functor category – Accessibility.

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.

A\phantom{A}(n,r)-categoriesA\phantom{A}A\phantom{A}toposesA\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

References

The term accessible category is due to

  • Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.1989.

The standard textbook on the theory of accessible categories is

See also

and

which further stratifies the accessible categories in terms of sound doctrines.

A discussion of accessible (∞,1)-categories is in section 5.4, p. 341 of

Some recent developments in the theory of accessible categories can be found in a series of papers on categorical model theory and abstract elementary classes (many of which also contain some results about arbitrary accessible categories), such as:

  • Tibor Beke, Jiří Rosický, Abstract elementary classes and accessible categories, 2011, arxiv

  • Michael Lieberman, Jiří Rosický, Sebastien Vasey, Internal sizes in μ-abstract elementary classes, arxiv

  • Michael Lieberman, Jiří Rosický, Sebastien Vasey , Set-theoretic aspects of accessible categories, arxiv

Last revised on July 29, 2019 at 18:11:38. See the history of this page for a list of all contributions to it.