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Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is there a known characterisation of the categories $\mathscr C$ that are:

  1. locally $\kappa$-presentable and cartesian-closed;
  2. locally $\kappa$-presentable and locally cartesian-closed;

in terms of being the free cocompletion of a small $\kappa$-cocomplete category with particular structure under $\kappa$-filtered colimits?

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  • $\begingroup$ I believe it should be possible to extract a characterisation of the locally cartesain closed locally presentable categories from Theorem 7.25 of Street's Cosmoi of internal categories, although note that footnote 2 ibid. seems in contradiction with mathoverflow.net/questions/294108/…. $\endgroup$
    – varkor
    Commented Sep 27 at 11:48
  • $\begingroup$ This issue is discussed in §4 of Borceux–Pedicchio's A characterization of quasi-toposes. $\endgroup$
    – varkor
    Commented Oct 1 at 11:27
  • $\begingroup$ A relevant reference for cartesian closure is Proposition 20 of Bastiani–Ehresmann's Categories of sketched structures. $\endgroup$
    – varkor
    Commented Oct 1 at 12:16
  • $\begingroup$ (I believe I have a characterisation of both conditions now. I will write up a proof later.) $\endgroup$
    – varkor
    Commented Oct 1 at 17:58

2 Answers 2

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I think this is the Day reflection theorem when viewing the LFP as a reflective subcategory of a presheaf topos. For the locally Cartesian closed case, I guess you would just apply some fibred category variant of the Day reflection theorem as each $$\mathcal{E}/X$$ is a reflective subcategory of the Cartesian closed $$[C,Set]/X$$.

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    $\begingroup$ This answer is coherent with Street's Thm. 3.11 in Cosmoi of internal categories. $\endgroup$
    – Ivan Di Liberti
    Commented Oct 24, 2020 at 8:09
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    $\begingroup$ Day's reflection theorem tells us that the LFP category $\mathscr C$ is cartesian-closed iff the reflector is cartesian: how can we characterise the corresponding category of finitely presentable objects for $\mathscr C$ using this? $\endgroup$
    – varkor
    Commented Oct 24, 2020 at 19:25
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I might try to improve my answer later this day. For the moment, a sufficient condition was given by Pedicchio and Borceux in A characterization of quasi-toposes, JoA 139 (1991).

Prop 4.1. If $C$ is an elementary topos, then $\mathsf{Lex}(C^\circ,\mathsf{Set})$ is locally cartesian closed.

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    $\begingroup$ Borceux and Pedicchio actually prove a stronger result: that if $C$ is a category with pullback-stable finite colimits, then $\mathrm{Lex}(C^{\text{op}}, \mathrm{Set})$ is locally cartesian closed. $\endgroup$
    – varkor
    Commented Oct 1 at 11:23

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