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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeSep 7th 2012

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. =–

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJul 31st 2019

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeMay 4th 2020

• Maru Sarazola, An introduction to locally finitely presentable categories, (pdf)
• CommentRowNumber4.
• CommentAuthorGuest
• CommentTimeJan 26th 2022
Is there an example of a category which is

- locally finitely presentable
- Barr-exact (aka effective regular)

but is not the category of algebras of a Lawvere algebraic theory?
• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJan 27th 2022
• (edited Jan 27th 2022)

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.)

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeAug 26th 2022

• Henning Krause, Functors on locally finitely presented additive categories, Colloq. Math. 75:1 (1998) pdf
• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeAug 26th 2022
• (edited Aug 26th 2022)

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 $C$:

• Francis Borceux, Carmen Quinteiro, A theory of enriched sheaves, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145–162 numdam
• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeAug 26th 2022
• (edited Aug 26th 2022)
• P. Gabriel, F. Ulmer, Lokal präsentierbare Kategorien, Springer Lect. Notes in Math. 221 1971 Zbl0225.18004 MR327863
• Jiří Adámek, Jiří Rosicky, Locally presentable and accessible categories, Cambridge University Press 1994.
• CommentRowNumber9.
• CommentAuthorvarkor
• CommentTimeJan 25th 2024