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Category theory; homological algebra Definition

Accessible category

Builds on CategoryFiltered categoryColimitRegular cardinalHom functor

Definition

Let λ\lambda be a regular cardinal. A small category ℐ\mathcal{I} is λ\lambda-filtered if every subcategory of ℐ\mathcal{I} with fewer than λ\lambda morphisms has a cocone in ℐ\mathcal{I}. Let 𝒞\mathcal{C} be a locally small category. An object AA of 𝒞\mathcal{C} is λ\lambda-presentable if the hom functor 𝒞(A,−)\mathcal{C}(A, -) preserves colimits of λ\lambda-filtered diagrams. The category 𝒞\mathcal{C} is λ\lambda-accessible if both conditions hold.

  1. Every diagram ℐ→𝒞\mathcal{I} \to \mathcal{C} with ℐ\mathcal{I} small and λ\lambda-filtered has a colimit in 𝒞\mathcal{C}.
  2. There is a set SS of λ\lambda-presentable objects such that every object of 𝒞\mathcal{C} is a colimit of a λ\lambda-filtered diagram of objects of SS.

The category 𝒞\mathcal{C} is accessible if it is λ\lambda-accessible for some regular cardinal λ\lambda.

Remarks

For λ=ℵ0\lambda = \aleph_0, the λ\lambda-filtered categories are the filtered categories, and one says finitely presentable and finitely accessible.

Example: the category of sets is finitely accessible. It has all colimits, so clause 1 holds. A finite set AA is finitely presentable, since a map from AA into a filtered colimit factors through some stage, essentially uniquely. Every set is the directed union, hence the filtered colimit, of its finite subsets, and the finite sets {1,…,n}\set{1, \dots, n} form a set SS up to isomorphism.

Non-example: let ℳ\mathcal{M} be the category with one object ∗* and morphisms 11 and ee, where e∘e=ee \circ e = e. The identity diagram of ℳ\mathcal{M} is λ\lambda-filtered for every λ\lambda, with the cocone ee. It has no colimit: the cocone ee is the only one, and both 11 and ee factor it through itself. So clause 1 fails for every λ\lambda, and ℳ\mathcal{M} is not accessible.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Category, concept, no evidence levelCategoryconceptColimit, concept, no evidence levelColimitconceptFiltered category, concept, no evidence levelFiltered categoryconceptHom functor, concept, no evidence levelHom functorconceptRegular cardinal, concept, no evidence levelRegular cardinalconceptAccessible category, definition, no evidence levelAccessible categorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Category concept
  2. Colimit concept
  3. Filtered category concept
  4. Hom functor concept
  5. Regular cardinal concept
  6. Accessible category definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Accessible_category
About this page
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Kinddefinition
WritingW0 stub
MSC18C35
DefinesAccessible category
UsesCategory, Filtered category, Colimit, Regular cardinal, Hom functor
Changed2 November 2026

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