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A well-known theorem states that a category E is a Grothendieck topos if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite limits.

In their paper Grothendieck quasitoposes, Garner and Lack proved that a category E is a Grothendieck quasitopos (the category of sheaves for one topology that are separated for another topology) if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector has stable units (i.e. preserves all pullbacks over objects in the subcategory) and preserves monomorphisms.

Is there a characterization of when a category E can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite products? (The reflector preserving finite products is equivalent to the reflective subcategory being an exponential ideal, i.e. $Y^X$ lies in the subcategory as soon as $Y$ does.) Of course, any such category must be locally presentable and cartesian closed. Can this be extended to a set of conditions which is both necessary and sufficient?

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In fact, the converse holds: every cartesian closed locally presentable category is a reflective subcategory of a presheaf category whose reflector preserves finite products. First, a locally presentable category is accessibly embedded in a presheaf category. Then, Theorem 3.10 of Street's Cosmoi of Internal Categories states that, given a reflective subcategory of a cartesian closed category, if the subcategory is cartesian closed and strongly generating, then the reflector preserves finite products. Clearly the latter property holds for locally presentable categories, as they are dense subcategories of presheaf categories.

(I must admit that I do not see why a reflective subcategory of a presheaf category whose reflector preserves finite products must be accessibly embedded; I would appreciate clarification on this point.)

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  • $\begingroup$ Better is true, no? Every cocomplete cartesian closed category is reflective in its category of small presheaves and the reflector preserves finite products (the argument is the same of Day's). Notice that finite products indeed do exists in the category of small presheaves by Remark 3.9 in Rosicky-Adamek "How nice are are free completions?" $\endgroup$
    – Ivan Di Liberti
    Commented Oct 2 at 18:51
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    $\begingroup$ @IvanDiLiberti: I don't see you obtain finite product preservation by the reflector in this case. Perhaps you could elaborate in a second answer? $\endgroup$
    – varkor
    Commented Oct 2 at 19:01
  • $\begingroup$ Sure. Let $A$ be a bicomplete cartesian closed category and consider the Yoneda embedding into small presheaves $y: A \to P(A)$. Now, the embedding preserves finite products and it is easy to see that because $A$ is cartesian closed, $A$ must be an exponential ideal in $P(A)$, indeed $y(c)^d = y(c)^{\text{colim} yc_i} = \text{lim} y(c)^{y(c_i)} = y(\text{lim} c^{c_i})$. So now we can apply Sketches A4.3.1. $\endgroup$
    – Ivan Di Liberti
    Commented Oct 2 at 22:02
  • $\begingroup$ The fact that $y(c)^y(a)$ is indeed $y(c^a)$ follows by direct inspection. Indeed $P(A)(F \times y(a), y(c)) = P(A)(\text{colim} y(d_i) \times y(a), y(c)) = \text{lim} P(A)( y(d_i) \times y(a), y(c)) = \text{lim} A( d_i \times a, c)...$. $\endgroup$
    – Ivan Di Liberti
    Commented Oct 2 at 22:20
  • $\begingroup$ Finally, I am perfectly aware that I added the hypothesis of completeness when answering your question. At the moment I can't find a sharper argument than this one. $\endgroup$
    – Ivan Di Liberti
    Commented Oct 2 at 22:21

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