6
$\begingroup$

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?

$\endgroup$

2 Answers 2

3
$\begingroup$

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 $$[C,Set]/X$$.

$\endgroup$
2
  • 1
    $\begingroup$ This answer is coherent with Street's Thm. 3.11 in Cosmoi of internal categories. $\endgroup$ Oct 24, 2020 at 8:09
  • 1
    $\begingroup$ 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, 2020 at 19:25
1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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, 2020 at 20:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.