nLab locally finitely presentable category





A locally finitely presentable category is an 0{}_0-locally presentable category.

We spell out what this means:

An object XX of a category CC is said to be finitely presentable (sometimes called compact or ‘finite’) if the representable functor C(X,)C(X,-) is finitary, i.e., preserves filtered colimits. Write C fpC_{fp} for the full subcategory of CC consisting of the finitely presentable objects.

A category CC satisfying (any of) the following equivalent conditions is said to be locally finitely presentable (or lfp):

  1. CC is the free cocompletion of a small finitely cocomplete category under filtered colimits: see ind-object.
  2. CC has all small colimits, the category C fpC_{fp} is essentially small, and any object in CC is a filtered colimit of the canonical diagram of finitely presentable objects mapping into it.
  3. CC is the category of models for an essentially algebraic theory. Here an ‘essentially algebraic theory’ is a small category DD with finite limits, and its category of ‘models’ is the category of finite-limit-preserving functors DSetD \to Set. (See Gabriel–Ulmer duality.)
  4. CC is the category of models for a finite limit sketch.
  5. C fpC_{fp} has finite colimits, and the restricted Yoneda embedding C[C fp op,Set]C\hookrightarrow [C_{fp}^{op},Set] identifies CC with the category of finite-limit-preserving functors C fp opSetC_{fp}^{op} \to Set.

Replacing “finite” by “of cardinality less than κ\kappa” everywhere, for some cardinal number κ\kappa, results in the notion of a locally presentable category.



  • P. Gabriel, F. Ulmer, Lokal präsentierbare Kategorien, Springer Lect. Notes in Math. 221 1971 Zbl0225.18004 MR327863
  • Jiří Adámek, Jiří Rosicky, Locally presentable and accessible categories, Cambridge University Press 1994.

Introductory account:

In additive context

  • Henning Krause, Functors on locally finitely presented additive categories, Colloq. Math. 75:1 (1998) pdf

If VV is a locally finitely presentable symmetric monoidal closed category then there is a bijection between exact VV-localizations of the VV-category of VV-valued VV-enriched presheaves on a VV-category CC and VV-enriched Grothendieck topologies on CC:

  • Francis Borceux, Carmen Quinteiro, A theory of enriched sheaves, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145–162 numdam MR1394507

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