nLab
accessible category

Idea

An accessible category is a possibly large category which is however essentially determined by a small category. There are several ways to make this precise, all equivalent.

Definition

A locally small category $C$ is accessible if for some regular cardinal $\kappa$:

• the category has $\kappa$-directed colimits (or, equivalently, $\kappa$-filtered colimits), and
• 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. Unlike for locally presentable categories, it does not follow that if $C$ is $\kappa$-presentable and $\kappa <\lambda$ then $C$ is also $\lambda$-presentable. It is true, however, that for any accessible category, there are arbitrarily large cardinals $\lambda$ such that $C$ is $\lambda$-presentable.

Equivalent characterizations include that $C$ is accessible iff:

• it is the category of models (in $\mathrm{Set}$) 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 important notion of functor between accessible categories is an accessible functor, meaning a functor $F:C\to D$ such that there exists a $\kappa$ such that $C$ and $D$ are both $\kappa$-accessible and $F$ preserves $\kappa$-filtered colimits.

Relation to other concepts

• If an accessible category in addition has all (small) colimits (or, equivalently, limits), then it is a locally presentable category.

• A small category is accessible precisely when it is idempotent complete. Makkai-Paré say that this means accessibility is an “almost pure smallness condition.”

Properties

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.

Proof

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 $\mathrm{Arr}(C)$.

Proof

This appears as HTT, corollary A.2.6.6.

In addition:

• The 2-category $\mathrm{Acc}$ of accessible categories, accessible functors, and natural transformations has all small 2-limits.

• 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).

References

The term accessible category is due to

• Makkai, Paré 1989.

The standard textbook on the theory of accessible categories is

• Adámek and Rosicky, Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge, 1994.

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

• Jacob Lurie, Higher Topos Theory