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. 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.”

• A functor out of an accessible category that preserves $\kappa$-filtered colimits for some regular cardinal $\kappa$ is an accessible functor.

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.

Proposition (closure under limits)

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

Proof

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

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 cardinal?s, then it has directed colimits of systems of faithful functors.

Proof

See Pare and Rosicky, Colimits of accessible categories.

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).

References

The term accessible category is due to

• Makkai, 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

• 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