A **locally finitely presentable category** is an ℵ${}_0$-locally presentable category.

We spell out what this means:

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

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

- $C$ is the free cocompletion of a small finitely cocomplete category under filtered colimits: see ind-object.
- $C$ 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 $C_{fp}^{op} \to Set$.

Replacing “finite” by “of cardinality less than $\kappa$” everywhere, for some cardinal number $\kappa$, 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:

- Maru Sarazola,
*An introduction to locally finitely presentable categories*(2017) [pdf, 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 numdam MR1394507

Last revised on June 7, 2024 at 06:52:02. See the history of this page for a list of all contributions to it.