nLab
accessible category

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 $C$ is accessible if for some regular cardinal $\kappa$:

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

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

If $C$ satisfies these properties for some $\kappa$, we say that it is $\kappa$-accessible.

Remark

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

Proposition

Equivalent characterizations include that $C$ is accessible iff:

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

• it is of the form ${\mathrm{Ind}}_{\kappa }(S)$ for $S$ small, i.e. the $\kappa$-ind-completion of a small category, for some $\kappa$.

• it is of the form $\kappa \mathrm{Flat}(S)$ for $S$ small and some $\kappa$, i.e. the category of $\kappa$-flat functors from some small category to $\mathrm{Set}$.

• it is the category of models (in $\mathrm{Set}$) of a suitable type of logical theory.

The relevant notion of functor between accessible categories is

Definition

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

Properties

Stability under various constructions

Proposition

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

(Lurie, prop. 5.4.4.3)

Proposition

(preservation of accessibility under inverse images)

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

This appears as HTT, corollary A.2.6.5.

Proposition

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

Let $C$ be a combinatorial model category, $\mathrm{Arr}(C)$ its arrow category, $W\subset \mathrm{Arr}(C)$ the full subcategory on the weak equivalences and $F\subset \mathrm{Arr}(C)$ the full subcategory on the fibrations. Then $F$, $W$ and $F\cap W$ are accessible subcategories of $\mathrm{Arr}(C)$.

This appears as HTT, corollary A.2.6.6.

Proposition

(closure under limits)

The 2-category $\mathrm{Acc}$ 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 $\mathrm{Acc}$ 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 $T$ is a theory of presheaf type precisely if its category $\mathrm{Mod}(T,\mathrm{Set})$ of models in Set is a finitely accessible category, and if and only if it is sketchable.

Related concepts

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

(n,r)-categories…	satisfying Giraud's axioms	inclusion of left exaxt localizations	generated under colimits from small objects		localization of free cocompletion		generated under filtered colimits from small objects
(0,1)-category theory	(0,1)-toposes	$\hookrightarrow$	algebraic lattices	$\simeq$ Porst’s theorem	subobject lattices in accessible reflective subcategories of presheaf categories
category theory	toposes	$\hookrightarrow$	locally presentable categories	$\simeq$ Adámek-Rosický’s theorem	accessible reflective subcategories of presheaf categories	$\hookrightarrow$	accessible categories
model category theory	model toposes	$\hookrightarrow$	combinatorial model categories	$\simeq$ Dugger’s theorem	left Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory	(∞,1)-toposes	$\hookrightarrow$	locally presentable (∞,1)-categories	$\simeq$ Simpson’s theorem	accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories	$\hookrightarrow$	accessible (∞,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

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

See also

• Robert Paré, Jiří Rosický, Colimits of accessible categories (arXiv:1110.0767)

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

• Jacob Lurie, Higher Topos Theory