@DenisNardin I second this. Adding the possibility of the refference discussing deformation of maps.

If it wasn't clear I was referring to the question whether anyone knows a reference for deformation of non commutative algebras.

@SaalHardali I suspect this is a typo, but for clarity's sake: I wasn't talking about noncommutative algebras (that's in HA and many other places), but nonconnective algebras

@CharlesRezk In fact, I think this might be the case. Let $E$ be a topos. Let $\kappa$ be large enough so that $E$ is $\kappa$-accessible and the $\kappa$-presentable objects are closed under finite limits. Then I think the standard presentation $E = Ind_\kappa(E_\kappa) \subseteq Psh(E_\kappa)$ might be given by a Grothendieck topology...

I guess I don't have a lot of evidence for this though

@TimCampion It's certainly a left-exact localization for essentially formal reasons, but I don't see any particular reason why that should be a topological localization.

For instance, suppose $E=\mathcal{S}$ ($\infty$-groupoids), and $\kappa=\omega$. Then you get an obvious left-exact localization $Psh(\mathcal{S}^\omega)\twoheadrightarrow \mathcal{S}$. What happens if you factor this as $Psh(\mathcal{S}^\omega) \to Sh(\mathcal{S}^\omega,T) \to \mathcal{S}$, where the first step is a topological localization and the second step inverts some $\infty$-connected maps?

I think that there are not very many interesting grothendieck topologies on $\mathcal{S}^\omega$, and so $T$ will actually be the trivial topology.

Okay, I really need a larger $\kappa$ there, since $\mathcal{S}^\omega$ is not closed under finite limits. But the same question stands for that.

And apparrently I don't mean the trivial topology, but rather the canonical topology.

So maybe that's not really an argument.

@MarcHoyois You're right. So this site has infinite cohomological dimension and yet everything is hypercomplete?

@CharlesRezk Yeah, I think I'd want to claim that for $\kappa$ sufficiently large, $E = Ind_\kappa(E_\kappa)$ coincides with sheaves in the canonical topology on $E_\kappa$. Still not clear why this should be so (it may well be false), although by HTT 6.2.4.6 we have one inclusion: $E$ is contained in the canonical-topology-sheaves.

Classically, sheaves on $E$ itself in the canonical topology coincide with $E$. Is this true $\infty$-categorically?

If so, you'd just have to cut down to a small subsite somehow.

I think that's basically the question. I don't know.

Let $F\colon \mathcal{S}^{\mathrm{op}}\to \mathcal{S}$ be a functor, and suppose it has the following property: for every map in $\mathcal{S}$ of the form $\coprod U_i\to X$ such that $\coprod \pi_0U_i\to \pi_0X$ is surjective, the evident map $F(X) \to \mathrm{lim}\bigl[ \prod F(U_i) \rightrightarrows \prod F(U_i\times_X U_j) \cdots \bigr]$ is an equivalence. Must $F$ be a representable functor?

I.e., if $F$ takes "effective colimits to limits", does it therefore take all colimits to limits?

Clearly $F$ takes all coproducts to products.

So it suffices to treat pushouts

$F$ takes pushouts of monomorphisms to pullbacks

The idea is that an $F$ which takes "effective colimits to limits" is exactly a sheaf on the canonical topology of $\mathcal{S}$.

But I'm not sure about pushouts of effective epimorphisms, or the pushout of an effective epimorphism along a mono

yeah

Why does $F$ take coproducts to products?

Because the coproduct inclusions form a covering sieve of monos

right?

Ok.

I guess (?) the condition on $F$ can be rephrased as: the map $F(\mathrm{colim}_{\mathcal{G}} X) \to \mathrm{lim}_{\mathcal{G}} F(X)$ is an equivalence for any functor $X\colon \mathcal{G}\to \mathcal{S}$ from a small $\infty$-groupoid $\mathcal{G}$.

No that's not right. -

@TomBachmann Right, in the end it's a presheaf ∞-topos, and these can have any dimension.

I guess this mainly shows that my intuition of cohomological dimension is completely wrong.

Thanks!

@CharlesRezk Actually -- I believed you when you said the canonical topology is characterized by the fact that a canonical sheaf preserves "effective colimits". But suddenly I'm not so sure -- I can't seem to find this in HTT 6.2.4. It seems to me we only know that in the canonical topology, a map of the form $\amalg_i X_i \to Y$, where $X_i \to Y$ is monic, has the colimit of its Cech nerve carried to a limit by any sheaf.

So maybe in the canonical topology there could still be effective epimorphisms such that sheaves don't satisfy descent along them?

I'm not really looking at Lurie's book here. I'm thinking of the following setup: $\mathcal{X}\xrightarrow{\rho} Psh(\mathcal{X}) \xrightarrow{a} \widehat{\mathcal{X}}$, where $\mathcal{X}$ is an $\infty$-topos, $\rho$ is Yoneda, and $a$ sends a functor to its colimit. The colimit will generally be a "large" object, hence the hat on the $\mathcal{X}$.

We have $a(\rho_X)\approx X$ for objects of $\mathcal{X}$. Since $\rho$ preserves finite limits, its a consequence that $a$ preserves finite limits. I am interested in the class of monomorphisms in $Psh(\mathcal{X})$ which $a$ sends to isomorphisms.

If we restrict to monomorphisms $F\rightarrowtail \rho_X$ whose target is a representable, this gives a Grothendieck topology $T$ on $\mathcal{X}$. This is what I am calling the "canonical topology".

A monomorphism $F\rightarrowtail \rho_X$ is the essential image of some family of maps $\{\rho_{U_i}\to \rho_X\}$, and as such is a colimit of a simplicial diagram $[n]\mapsto \coprod \rho_{U_{i_0}\times_X\dots \times_X U_{i_n}}$ in $Psh(\mathcal{X})$.

I'm dancing around the fact that objects of $PSh(\mathcal{X})$ are not generally small colimits of representables, but this is supposed to be fixed by replacing with presheaves on a suitable small category $\mathcal{X}_\kappa\subseteq \mathcal{X}$.

So $F\rightarrowtail \rho_X$ coming from $\{\rho_{U_i}\to \rho_X\}$ is in $T$ iff $X\leftarrow \mathrm{colim}\bigl[ [n]\mapsto \coprod U_{i_0}\times_X\cdots\times_X U_{i_n} \bigr]$ is an equivalence in $\mathcal{X}$.

I really want to replace $PSh(\mathcal{X})$ by the full subcategory $PSh_s(\mathcal{X})$ spanned by functors which are small colimits of representables. Then everything seems to make sense as stated.