# What is a morphism of ∞-sites?

Recall that a morphism of sites is a covering-flat functor that preserves covering families.

Morphisms of sites can be identified with those geometric morphisms of induced toposes for which the inverse image functor preserves representables.

If both sites have finite limits, then covering-flat functors are precisely the functors that preserve finite limits.

What is the analogue of this story for ∞-sites?

Again, we want to identify morphisms of ∞-sites with those geometric morphisms of induced (nonhypercompleted) ∞-toposes for which the inverse image functor preserves representables.

Thus, a morphism of ∞-sites should be a functor between the underlying ∞-categories that preserves covering families and satisfies the appropriate ∞-analogue of being a covering-flat functor.

What is a covering-flat functor between ∞-sites?

If both ∞-sites admit finite ∞-limits, then covering-flat functors could be defined as finite ∞-limit preserving functors.

However, I am interested in ∞-sites that do not have finite ∞-limits, such as the site of smooth manifolds or the site of cartesian spaces.

• I havn't really thought about it, but you might get two different answer depending if you use site to represente hypercomplete $\infty$-topos or topological localization of presheaves categories... – Simon Henry Mar 21 at 0:39
• @SimonHenry: The toposes in this question are not hypercompleted, although an answer in the hypercompleted case would also be interesting. – Dmitri Pavlov Mar 21 at 0:48
• the appendix of this paper develops some of this theory (but maybe not in the generality you wanted?): arxiv.org/abs/1803.01804 – Dylan Wilson Mar 21 at 1:12
• @DylanWilson: Definition A.10 of morphisms of ∞-sites in this paper assumes the existence of pullbacks, i.e., finite ∞-limits. Also, Proposition A.13 constructs geometric morphisms of ∞-toposes under a rather strong additional assumption of a covering lifting property. – Dmitri Pavlov Mar 21 at 1:32
• @DmitriPavlov that's why I said it might not be the generality you want. but not all pullbacks are assumed, only pullbacks along covers. so the site of smooth manifolds is okay here, right? anyway, I agree the theory in that appendix is probably not quite what you want, and only mention it because I thought it may still be helpful. – Dylan Wilson Mar 21 at 11:56