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

Let $\kappa$ be a regular cardinal. Recall an object $X: \mathcal{C}$ is $\kappa$-compact iff $\mathcal{C}(X,-)$ commutes with $\kappa$-filtered colimits.

Definition

A locally small category $\mathcal{C}$ is $\kappa$-accessible if:

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.

Then $\mathcal{C}$ is an accessible category if there exists a $\kappa$ such that it is $\kappa$-accessible.

Remark

Unlike for locally presentable categories, it does not follow that if $\mathcal{C}$ is $\kappa$-accessible and $\kappa\lt \lambda$ then $\mathcal{C}$ 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 $Set$.

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

The relevant notion of functor between accessible categories is

Definition

A functor $F\colon C\to D$ 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

Raising the index of accessibility

If $\mathcal{C}$ is $\lambda$-accessible and $\lambda\unlhd\mu$ (see sharply smaller cardinal), then $\mathcal{C}$ is $\mu$-accessible. Thus, any accessible category is $\mu$-accessible for arbitrarily large cardinals $\mu$.

Stability under various constructions

Proposition

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

(Lurie, prop. 5.4.4.3)

Proposition

(preservation of accessibility under inverse images)

Let $F : \mathcal{C} \to \mathcal{D}$ be a functor between locally presentable categories which preserves $\kappa$-filtered colimits, and let $\mathcal{D}_0 \subset \mathcal{D}$ be an accessible subcategory. Then the inverse image $f^{-1}(\mathcal{D}_0) \subset C$ 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 $\mathcal{C}$ be a combinatorial model category, $Arr(\mathcal{C})$ its arrow category, $W \subset Arr(\mathcal{C})$ the full subcategory on the weak equivalences and $F \subset Arr(\mathcal{C})$ the full subcategory on the fibrations. Then $F$, $W$ and $F \cap W$ are accessible subcategories of $Arr(\mathcal{C})$.

This appears as HTT, corollary A.2.6.6.

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 that are complete and cocomplete (i.e. are locally presentable): a functor between such categories 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 (1989) say that this means accessibility is an “almost pure smallness condition.”

Categories of models over a theory

Proposition

For a category $\mathcal{M}$ the following are equivalent:

• $\mathcal{M}$ is finitely accessible.

• $\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set)$ for some theory $\mathbb{T}$ of presheaf type.

Moreover, one has the following result due to Christian Lair:

Proposition

For a category $\mathcal{M}$ the following are equivalent:

• $\mathcal{M}$ is accessible.

• $\mathcal{M}$ is sketchable.

See also at categorical model theory.

Well-poweredness and well-copoweredness

• Every accessible category $\mathcal{C}$ is well-powered, since it has a small dense subcategory $\mathcal{A}$, for which the restricted Yoneda embedding $\mathcal{C} \to [\mathcal{A}^{op},Set]$ is fully faithful and preserves monomorphisms, hence embeds the subobject posets of $\mathcal{C}$ as sub-posets of those of $[\mathcal{A}^{op},Set]$.

• Every accessible category with pushouts is well-copowered. This is shown in Adamek-Rosicky, Proposition 1.57 and Theorem 2.49. Whether this is true for all accessible categories depends on what large cardinal properties hold: by Corollary 6.8 of Adamek-Rosicky, if Vopenka's principle holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large measurable cardinals.

The 2-category of accessible categories

Write AccCat for the 2-category whose

• objects are accessible categories,

• 1-morphisms are accessible functors

• 2-morphisms are natural transformations.

Proposition

The 2-category AccCat has all (lax) 2-limits and these are preserved by the inclusion $AccCat \to$ Cat.

This appears as Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8, Adámek & Rosický (1994) around Thm. 2.77.

Proposition

Given a cosmos for enrichment $\mathcal{V}$ which is (symmetric monoidal closed and) locally presentable, then the 2-category $\mathcal{V}$-AccCat of $\mathcal{V}$-enriched accessible categories has all PIE 2-limits and splittings of idempotent equivalences (equivalently it has all flexible 2-limits), as well as 2-pullbacks along isofibrations.

The analogous statements holds for $\mathcal{V}$-enriched and conically accessible categories, in which case the forgetful functor $\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat$ preserves these 2-limits.

This is Lack & Tendas (2023), Thm. 5.5, Thm. 5.9.

Proposition

(directed unions)
The 2-category AccCat has directed 2-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.

Paré & Rosický (2013)

Examples

• Every small discrete category is $\kappa$-accessible for every regular cardinal $\kappa$, since every discrete filtered diagram is trivial.

Functor categories

See at Functor category – Accessibility.

Related concepts

• class-accessible category

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.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

General

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, (1989) [ISBN:978-0-8218-7692-3]

Further monographs (with focus on locally presentable categories):

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189, Cambridge University Press (1994) [doi:10.1017/CBO9780511600579]

See also

• Robert Paré, Jiří Rosický, Colimits of accessible categories, Math. Proc. Cambr. Phil. Soc. 155 (2013) 47-50 [doi:10.1017/S0305004113000030, arXiv:1110.0767]

• Jiří Adámek, Francis Borceux, Stephen Lack, Jiří Rosický, A classification of accessible categories, Journal of Pure and Applied Algebra 175:7-30, 2002. abstract

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

The concept is studied in a 2-categorical setting in

• John Bourke, Accessible aspects of 2-category theory , arXiv:2003.06375 (2020). (abstract)

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

• Jacob Lurie, Higher Topos Theory

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

See also:

• Charles Rezk, Generalizing accessible ∞-categories, 2021 (pdf)

In enriched category theory

Discussion of accessible enriched categories (see also references on enriched locally presentable categories):

• Francis Borceux, Carmen Quinteriro, Enriched accessible categories, Bull. Austral. Math. Soc. 54 (1996) 489-501 [doi:10.1017/S0004972700021900]

• Francis Borceux, Carmen Quinteriro, Jiří Rosický, A theory of enriched sketches, Theory and Applications of Categories, 4 3 (1998) 47-72 [tac:4-03, pdf]

• Stephen Lack, Giacomo Tendas, Virtual concepts in the theory of accessible categories, Journal of Pure and Applied Algebra 227 2 (2023) 107196 [arXiv:10.1016/j.jpaa.2022.107196, arXiv:2205.11056]

  New characterizations of (enriched) accessible categories and introducing the notions of virtual orthogonality and virtual reflectivity.