nLab
locally finitely presentable category

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

Mike: Do people really call finitely presentable objects “finitary”? I’ve only seen that word applied to functors (those that preserve filtered colimits).

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 a finitary essentially algebraic theory.

   Does this essentially algebraic theory also have to be finitary?; that is, if it's an algebraic theory, then it's a Lawvere theory? —Toby

   Mike: Yes, it certainly has to be finitary. Possibly the standard meaning of “essentially algebraic” implies finitarity, though, I don’t know.

   Toby: I wouldn't use ‘algebraic’ that way; see algebraic theory.

3. $C$ is equivalent to the category of finite-limit-preserving functors $D\to \mathrm{Set}$ for some small category $D$ with finite limits.

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 $C_{\mathrm{fp}}^{\mathrm{op}}\to \mathrm{Set}$.

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

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.