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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 κ\kappa-limits where κ\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

    Added mention of Gabriel–Ulmer duality.

    diff, v24, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 4th 2020

    Added to references

    • Maru Sarazola, An introduction to locally finitely presentable categories, (pdf)

    diff, v25, current

    • 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][C^{op}, Set], this is equivalent to the category of lex functors DSetD \to Set where D opD^{op} is the finite-colimit completion of CC, if I have my variances straight.)

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeAug 26th 2022

    In additive context

    • Henning Krause, Functors on locally finitely presented additive categories, Colloq. Math. 75:1 (1998) pdf

    diff, v26, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeAug 26th 2022
    • (edited Aug 26th 2022)

    If VV is a locally finitely presentable symmetric monoidal closed category then there is a bijection between exact localizations of the VV-category of VV-enriched presheaves on a VV-category CC and enriched Grothendieck topologies on CC:

    • Francis Borceux, Carmen Quinteiro, A theory of enriched sheaves, Cahiers Topologie Géom. Différentielle Catég. 37 (1996), no. 2, 145–162 numdam

    diff, v26, current

    • 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.

    diff, v26, current

    • CommentRowNumber9.
    • CommentAuthorvarkor
    • CommentTimeJan 25th 2024
    • CommentRowNumber10.
    • CommentAuthorvarkor
    • CommentTimeJun 7th 2024

    Made explicit the characterisation in terms of ind-objects.

    diff, v29, current