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I’m making some edits to locally finitely presentable category, and removing some old query boxes. A punchline was extracted, I believe, from the first query box. The second I don’t think is too important (it looks like John misunderstood).
+–{: .query} 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. =–
+– {: .query} 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. =–
Added mention of Gabriel–Ulmer duality.
Almost any presheaf topos would do. (A topos is Barr-exact, and in the presheaf case $[C^{op}, Set]$, this is equivalent to the category of lex functors $D \to Set$ where $D^{op}$ is the finite-colimit completion of $C$, if I have my variances straight.)
In additive context
If $V$ is a locally finitely presentable symmetric monoidal closed category then there is a bijection between exact localizations of the $V$-category of $V$-enriched presheaves on a $V$-category $C$ and enriched Grothendieck topologies on $C$:
Added a link to locally strongly finitely presentable category.
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