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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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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 Oct 24 '20 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 Oct 24 '20 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 Borcerux 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 cartesian closed.

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  • $\begingroup$ Thank you; I was unaware of this reference. Just a note that you're missing "locally" in the statement of Prop. 4.1. $\endgroup$ – varkor Oct 24 '20 at 20:03

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