A "small" definition of sub-(∞,1)-topoi

Question

Suppose I have an $(\infty,1)$-topos $\mathcal{X}$ and a (small) set of maps $S$ in $\mathcal{X}$, which therefore generates an accessible localization $S^{-1}\mathcal{X}$. Is there any "small" condition $P$ on $S$ which implies that this localization is left exact (hence a sub-$(\infty,1)$-topos), and such that any accessible left exact localization is generated by some $S$ satisfying condition $P$?

By "small" I mean expressible with only (small-set-)bounded quantification and reference to categorical operations. So, for instance, "the localization $S^{-1}\mathcal{X}$ is left exact" is not small because it is a statement about all pullbacks in $\mathcal{X}$, which form a proper class (or a set in the next higher universe, whatever). The condition of 6.2.1.1(b) in Higher Topos Theory is likewise not small.

But, for instance, "$S$ consists of monomorphisms" is a small condition, since it quantifies only over the small set $S$. (Being a monomorphism is a small condition in the sense I mean; the ordinary definition of "monomorphism" quantifies over all objects of the category, but it's equivalent to the diagonal being an isomorphism, and the diagonal is a "categorical operation". I hope the intent is clear; if anyone is confused I could try to formulate a more precise definition of "smallness".)

Of course, "$S$ consists of monomorphisms" is probably not an answer to the question. I suspect that there is a small condition "$S$ consists of monomorphisms and ..." which answers a modified version of the question that asks about topological localizations (although at the moment I don't even see how to prove that). But I would really like an answer applying to all accessible left exact localizations.

Answer 1

This follow from some recent (I heard about this a year ago) results by Anel, Biedermann, Finster and Joyal.

Unfortunately their work is not available yet, but You have some slide of Mathieu Anel on the topic on his web page presenting this (http://mathieu.anel.free.fr/mat/doc/Anel-LexLocalizations.pdf). I don't know if other references about this are available.

What Anel's slide says is that given a set of arrows $S$ in an $\infty$-topos the smallest left exact localization inverting $S$ is the cocontinuous localization inverting all pullbacks of all iterated diagonal of maps in $S$.

Another way to say it is that a localization is left exact if and only if the class of all morphisms inverted is stable under pullback and taking diagonal... Which is actually also equivalent to that fact that has a full subcategory of $\mathcal{X}^{\Delta[1]}$ it is closed under finite limits. I have not seen their proof yet, but when it is formulated this way it is not too hard to give a very direct proof.

Using the descent property and the pullback stability of this class, it is enough to invert all such arrows whose target belong to some generating familly of the topos, so it provides the kind of small condition you are looking for.

For example, Given $\mathcal{X}$ an $\infty$-topos, $(x_i)$ a generating set of $\mathcal{X}$ (of special interest, $\mathcal{X}$ is a presheaf topos and $x_i$ are the representable), then left exact localization of $\mathcal{X}$ are all obtained as localization at a set $S$ of morphisms such that:

• every morphisms in $S$ has one of the $x_i$ as its target.
• Every pullback of an iterated diagonal of $s \in S$ whose target is one of the $x_i$ is in $S$, i.e. for all $f : X \rightarrow x_i \in S$, and for all maps $x_j$ to the target of the $n$-iterated diagonal of $f$, the pullback of the $n$-th iterated diagonal of $f$ to $x_j$ is in $S$.

In the case of topological localization, i.e. when all maps in $S$ are mono, $n$-iterated diagonal for $n \geqslant 1$ are all iso, so the thing about diagonal disapear and you recover the usual pulblack stability axiom for Grothendieck topology.