A sheaf is a presheaf that preserves small limits

Question

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.

However when reading Lurie's DAGVII he goes on to define a sheaf of spectra on an $\infty$-topos $\mathfrak X$ as a presheaf $\mathcal O:{\mathfrak X}^{op}\to \mathsf {Sp}$ which preserves small limits.

Why can the higher analogue of sheaves of rings be defined like this? My guess is that, because it is higher, it sorts out whatever problems you get when defining a normal sheaf like that. But I am seriously clueless on this matter and I would love for some helpful explanations.

Answer 1

This has nothing to do with $\infty$-categories, but with the fact that we look at the full topos and not an arbitrary site of definition:

Theorem: If $T$ is a (Grothendieck) ($1$-)topos, then a "sheaf of set" on $T$ is the same as a functor from $T^{op}$ to $Set$ sending colimits in $T$ to limits in Sets.

Sketches of Proof: Sheaf of sets on T, mean representable so they clearly satisifes this condition. Conversely, if something satisfies this conditons then if you restrict it to a site of definition of T, you get a sheaf on this site, you can then check that your functor is isomorphic to the representable at this sheaf.

You can replace set by group, or any reasonable category (it has to have limits and be well behaved enough so that sheafication exists...) Also this kind of theorem is not true just for $T$ a topos, but in fact for $T$ any presentable category. (this follows from the special adjoint functor theorem... although I think its more like a lemma that you need to prove the special adjoint functor theorem)

But this of course does not work if you replace $T$ by some arbitrary site of definition of $T$.