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 $C_{\mathrm{fp}}$ 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. $C$ has all small colimits, the category $C_{\mathrm{fp}}$ is essentially small, and any object in $C$ is a filtered colimit of the canonical diagram of finitely presentable objects mapping into it.
2. $C$ 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 $D\to \mathrm{Set}$.
3. $C$ is the category of models for a finite limit sketch.
4. $C_{\mathrm{fp}}$ has finite colimits, and the restricted Yoneda embedding $C\hookrightarrow [C_{\mathrm{fp}}^{\mathrm{op}},\mathrm{Set}]$ identifies $C$ with the category of finite-limit-preserving functors $C_{\mathrm{fp}}^{\mathrm{op}}\to \mathrm{Set}$.

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.