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

$C$ has all small colimits, the category $C_{fp}$ is essentially small, and any object in $C$ is a filtered colimit of the canonical diagram of finitely presentable objects mapping into it.

$C$ is the category of models for an essentially algebraic theory. Here an ‘essentially algebraic theory’ is a small category $D$ with finite limits, and its category of ‘models’ is the category of finite-limit-preserving functors $D \to Set$. (See Gabriel–Ulmer duality.)

$C$ is the category of models for a finite limit sketch.

$C_{fp}$ has finite colimits, and the restricted Yoneda embedding$C\hookrightarrow [C_{fp}^{op},Set]$ identifies $C$ with the category of finite-limit-preserving functors $C_{fp}^{op} \to Set$.

P. Gabriel, F. Ulmer, Lokal präsentierbare Kategorien, Springer Lect. Notes in Math. 221 1971 Zbl0225.18004MR327863

Jiří Adámek, Jiří Rosicky, Locally presentable and accessible categories, Cambridge University Press 1994.

A good introductory account is in

Maru Sarazola, An introduction to locally finitely presentable categories, (pdf)

In additive context

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

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

Francis Borceux, Carmen Quinteiro, A theory of enriched sheaves, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145–162 numdamMR1394507

Last revised on August 26, 2022 at 23:04:31.
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