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In a Grothendieck $\infty$-topos, it is known that, for arbitrarily large regular cardinals $\kappa$, there is a classifier for the class of relatively $\kappa$-compact morphisms. It is also easy to show that this is not the case in 1-toposes, because we might have isomorphic, but not equal, such morphisms classified by the same map. However, we should be able to recover at least part of the definition of an object classifier. Namely, I need to know that in a Grothendieck 1-topos, for arbitrarily large regular cardinals $\kappa$, there is a map $t: U' \to U$ such that for every relatively $\kappa$-compact morphism $f: X \to Y$ there exists a pullback square

$\require{AMScd}$ \begin{CD} X @>>> U'\\ @VfVV @VVtV\\ Y @>>> U \end{CD}

(not necessarily unique and such that the map $Y \to U$ doesn't necessarily only classify $f$). I feel that this should definitely be true, but I can't find it anywhere in the literature. It would be very nice to have a reference for it, or a confutation in the unfortunate case I'm wrong. Thanks!

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    $\begingroup$ There is a standard strategy that always works. Say that your topos is $\mathsf{Sh}(C)$ for a site $C$. Now, if such an object $U$ exists, it means that $$U(c) \stackrel{\text{Yon}}{\cong} \text{Nat}(y(c), U) \stackrel{\text{U.P.}}{\cong} \{\kappa\text{-compact morphism over } c\}.$$ This observation gives you the only possible definition for $U$ and is the standard strategy to build the sub. classifier. Whether this is a sheaf, I don't know. $\endgroup$
    – Ivan Di Liberti
    Commented Sep 23, 2020 at 11:03
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    $\begingroup$ Your second step is not applicable to my case, because there might be more than one relatively $\kappa$-compact morphism over $y(c)$ corresponding to a map $y(c) \to U$. Of course all the candidates will only differ by an isomorphism, but still it doesn't give a bijection of sets. That's the main issue in moving from $\infty$- to 1-toposes. $\endgroup$
    – Giulio Lo Monaco
    Commented Sep 23, 2020 at 11:39
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    $\begingroup$ A weak classifier of k-compact families can be obtained in a 1 topos in a standard way due to hofmann and streicher. For presheaves, there is a strict enough way to give a correct definition of a presheaf in the spirit of Ivan’s attempt (which sadly did not have a functorial action). Streicher observed that by sheafifying this weakly classifying family you get a weakly classifying family for sheaves too! this follows in essence from the left exactness of the localization . If people want me to put the details into an answer, i can try to do so. $\endgroup$
    – Jonathan Sterling
    Commented Oct 2, 2020 at 1:59
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    $\begingroup$ @JonathanSterling I worked out a proof myself in the meantime, which is indeed in the spirit of Ivan's comment. On presheaves, it is based on choosing a set of representatives for equivalence classes of relatively $\kappa$-compact morphisms over $y(c)$ as elements of $U(c)$. This can be transferred through a left exact localization, provided that we are not asking for uniqueness of the classifying maps. Is the strategy you have in mind similar to this? $\endgroup$
    – Giulio Lo Monaco
    Commented Oct 2, 2020 at 13:27
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    $\begingroup$ @GiulioLoMonaco This is the standard reference for the construction I referred to: www2.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf If yours turns out to be different, it might be a nice result that type theorists would appreciate. I encourage you to write it up in an answer. $\endgroup$
    – Jonathan Sterling
    Commented Oct 2, 2020 at 21:13

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If $\mathcal{X}$ is a topos, then there is an adjunction $\mathcal{X} \leftrightarrows \mathcal{P(C)}$ with a category of presheaves of sets, where the right adjoint is fully faithful and accessible and the left is left exact. The idea is very similar to the one spelled out in https://www2.mathematik.tu-darmstadt.de/~streicher/NOTES/lift.pdf, except that where its authors first find a suitable class of relatively $\kappa$-compact morphisms in $\mathcal{P(C)}$ and then consider the class of all morphisms in $\mathcal{X}$ that become relatively $\kappa$-compact in $\mathcal{P(C)}$, my analysis allows to find some $\kappa$ such that the desired class is that of all relatively $\kappa$-compact morphisms in $\mathcal{X}$. A complete proof can be found in Appendix A of http://www.tac.mta.ca/tac/volumes/37/5/37-05.pdf

I'd like to point out that neither strategy is more general than the other. The reason of this is that a class of relatively $\kappa$-compact morphisms in $\mathcal{X}$ need not be sent precisely to a class of relatively $\lambda$-compct morphisms in $\mathcal{P(C)}$ and, conversely, the preimage of a class of relatively $\lambda$-compact morphisms in $\mathcal{P(C)}$ need not be precisely a class of relatively $\kappa$-compact morphisms in $\mathcal{X}$.

However, all relatively $\kappa$-compact morphisms found in the latter proof are indeed sent to relatively $\kappa$-compact morphisms in $\mathcal{P(C)}$.

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