This question on MO made me realize that I don’t even know the right definition of surjective geometric morphism for $(\infty,1)$-toposes. For 1-toposes, it is equivalent for $f^*$ to be (1) comonadic, (2) conservative, and (3) faithful. Of course (2)⇔(3) doesn’t categorify (what does faithfulness even mean for (∞,1)-categories?). But I don’t see that (1)⇔(2) categorifies either, because the (∞,1)-comonadicity theorem (unlike the 1-categorical one) is not about *finite* limits, but $f^*$ is still only assumed to preserve finite limits.

Of course we want a (surjection, inclusion) factorization. For that, it seems almost certain that $f^*$ being conservative is the right definition, at least if we take the obvious definition of “inclusion” as $f_*$ being fully faithful. We can then produce the desired factorization of $f: \mathcal{F} \to \mathcal{E}$ by localizing $\mathcal{E}$ at the class of morphisms inverted by $f^*$.

But what happens if instead we construct the category of coalgebras of the comonad $f^* f_*$? Is the category of coalgebras of an (accessible) lex comonad on an $(\infty,1)$-topos again an $(\infty,1)$-topos? If so, what sort of factorization does this produce?

]]>If generalized nonabelian cohomology, from the nPOV, means a hom-space in some $(\infty,1)$-topos, then it can equivalently be characterized as global sections of an object in some $(\infty,1)$-topos, since for any $X,A$ in an $(\infty,1)$-topos $E$ we have $Hom_E(X,A) = Hom_{E/X}(1,X^*A) = \Gamma_{E/X}(X^*A)$. Recall that traditional “abelian” sheaf cohomology $H^n(X;A)$ is the case when $A$ is an $n$-fold delooping of a discrete abelian group object, and when $A$ is locally constant (whatever that means) it reduces to “cohomology with local coefficients” and further to the most traditional algebraic-topology sort of cohomology when $A$ is constant. Generalizing in a different direction, if $A$ is constant, not on a discrete abelian group, but on a spectrum, then we obtain classical “generalized cohomology”, and if we further generalize to spectrum objects in $E$ then we have “generalized sheaf cohomology” with coefficients in a sheaf of spectra. Note that $\Gamma$ preserves abelian group objects and spectrum objects, so that with these definitions, abelian cohomology theories always produce abelian objects.

In another thread I asked a question about *homology* from the nPOV (and David C kindly supplied some links to past discussions). A couple of answers were given, but I just thought of a slightly different way of stating it, which I like. Suppose our $(\infty,1)$-topos is locally ∞-connected, so that in addition to a right adjoint $\Gamma$, the constant stack functor $\Delta$ has a left adjoint $\Pi$. Now $\Pi$ won’t preserve abelian and spectrum objects, but by general “adjoint lifting theorems” I would nevertheless expect to be able to *construct* from it a functor $\Pi_{spectra}: Spectra(E) \to Spectra$ which is left adjoint to $\Delta_{spectra}: Spectra \to Spectra(E)$. It seems to me that it would make sense to regard $\Pi_{spectra}$ as a notion of “sheaf homology” with coefficients in a sheaf of spectra (perhaps a constant one).

It could be that this is way off-base, but I’m getting my intuition from the May-Sigurdsson theory of “parametrized” spectra, which should morally (I believe) be identifiable with “locally constant” sheaves of spectra over nice spaces. In their context, the pullback functor $r^* : Spectra \to Spectra/X$ always has both a left adjoint $r_!$ and a right adjoint $r_*$, and the left adjoint is homology while the right adjoint is cohomology. In particular, $r_! r^* M$, for a spectrum $M$, can be identified with the generalized homology theory $H_*(X;M)$.

Thoughts? Is this obviously true? Obviously false?

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