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).

Toby: I have heard ‘finite’; see compact object.

Mike: Yes, I’ve heard ‘finite’ too.

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.

Toby: In the list of equivalent conditions above, does this essentially algebraic theory also have to be finitary?; that is, if it's an algebraic theory, then it's a Lawvere theory?

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.

John Baez: How come the first sentence of this paper seems to suggest that the category of models of any essentially algebraic theory is locally finitely presentable? The characterization below, which I did not write, seems to agree. Here there is no restriction that the theory be finitary. Does this contradict what Mike is saying, or am I just confused?

Mike: The syntactic category of a non-finitary essentially algebraic theory is not a category with finite limits but a category with $\kappa$-limits where $\kappa$ is the arity of the theory. A finitary theory can have infinitely many sorts and operations; what makes it finitary is that each operation only takes finitely many inputs, hence can be characterized by an arrow whose domain is a finite limit. I think this makes the first sentence of that paper completely consistent with what I’m saying.

Examples

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

• Top, FinSet are not lfp.