An object of a category is said to be finitely presentable (or finitary, sometimes called compact) if the representable functor preserves filtered colimits. Write for the full subcategory of 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 satisfying (any of) the following equivalent conditions is said to be locally finitely presentable (or lfp):
Replacing “finite” by “of cardinality less than ” everywhere, for some cardinal number , 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 -limits where 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.