Yonatan Harpaz

@DenisNardin, are you here?

Denis are you here?

I call it projective because in this case Sh(C) is a retract of the free presentable oo-category PSh(C) in Pr^L. In fact, in this case both Sh(C) and coSh(C) will be projective (hence both will be oo-topoi) and one can show that Sh(C) = LFun(coSh(C),Spaces).

The only case where I know how to show this is the very special "projective" case. This is the case where the localization PSh(C) -> Sh(C) is not just left exact, but actually preserves all limits. This happens, for example, if every object has an initial covering sieve (for example, if you take a compact Hausdorff space X and consider a sieve on U to be covering if it contains all V -> U in which the closure of V belongs to U).

@SaalHardali, I think this perspective is correct. In particular, suppose that your oo-topos is sheaves on some site C. Then indeed LFun(Sh(C),Spaces) = coSh(C), and so if Sh(C) is dualizable then its dual should be coSh(C). However, to have duality you would also need to have that Sh(C) = Fun(coSh(C),Spaces), and I'm not sure how to show that (or even if it's true in this generality).

see also the answer to mathoverflow.net/questions/252698/…

'm pretty sure they are dual to each other. Also one can actually prove that (unless X is trivial), these $\infty$-categories are actually not presheaf categories.

For example, take a compact Hausdorff space X, and for each U in X, consider the sieve on U consisting of those V \subseteq U such that the closure of V is contained in U. This collection of sieves doesn't satisfy the axioms of a Grothendieck topology, but regardless of that we can still talk about sheaves and cosheaves of spaces with respect to these sieves. In this case the $\infty$-category of sheaves is a retract of P(O(X)) and the category of cosheaves is a retract of P(O(X)^{op}), and I

@SaalHardali, Presheaf $\infty$-categories are dualizable, where the dual of P(C) is P(C^{op}). After that there are also presentable $\infty$-categories which are retracts of presheaf $\infty$-categories, i.e., they are accessible localizations D \subseteq P(C) such that the inclusion is both a left and a right functor.

@TomerSchlank, the derived mapping spectrum can be computed from the derived mapping space by taking a spectrum object $Y_{\bullet}$ in M such that $Y_0 = Y$ and then taking the spectrum $Map^h(X,Y_{\bullet})$. Your question can then be reduced to mapping spaces. Let $\mathbb{Z}$ be the integers considered as a complex concentrated in degree $0$. Then the underlying space of the mapping complex $Hom_M(X,Y)$ is $Map^h(\mathbb{Z}, Hom_M(X,Y)) \simeq Map^h(X,Y)$ by adjunction.

@QiaochuYuan, @AaronMazel-Gee. It's exactly what you said. As a covariant functor from $Cat$ to $Pr^L$ the functor $Psh(-)$ preserves small colimits, which is equivalent to saying that as a contravariant functor from $Cat$ to $Pr^R \cong (Pr^L)^{op}$ it takes small colimits to small limits. Since the forgetful functor $Pr^R \to \widehat{Cat}$ preserves small limits you can also think of $Psh(-)$ as a contravariant functor from $Cat$ to $\widehat{Cat}$ which take small colimits to small limits.