nLab
accessible category

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 accessible if for some regular cardinal κ\kappa:

  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.

If CC satisfies these properties for some κ\kappa, we say 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

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).

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.

Examples

Functor categories

See at Functor category – Accessibility.

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 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

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

Accessible categories in the context of categorical model theory are further discussed in

Revised on September 1, 2014 08:12:45 by Urs Schreiber (82.113.98.44)