nLab locally finitely presentable category

Definition

Definition

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):

1. $C$ is the free cocompletion of a small finitely cocomplete category under filtered colimits: see ind-object.
2. $C$ 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.
3. $C$ 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.)
4. $C$ is the category of models for a finite limit sketch.
5. $C_{fp}$ 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.

Examples

• Set, Graph, Pos, Cat, Ab are all lfp.

• Top, FinSet are not lfp.

Related pages

• locally strongly finitely presentable category

References

• 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