trans-categorical descent

General Paradigm

A morphism of geometric objects $f:X\to Y$ induces a map of categorification towers (Defn - omega-monoids) , with components $n{f}^{\ast }:nQCoh(Y)\to nQCoh(X)$ We’d like to compare

• $n$-categorical descent data along $n{f}^{\ast }$, with
• $(n+1)$-categorical descent data along on $(n+1)f^{\ast }$.

Case study: $E_{\infty }$-rings

Let $\epsilon :R\to \mathbb{S}$ be an augmented $E_{\infty }$, let $Spf(R)$ denote the corresponding pointed presheaf. ($Spf$ here is not used here to denote any notion of smallness or completion, but rather to evoke the feeling of working in a setting with pointed affines).

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Lemma Writing $X:=Spf(R)$, we have an equivalence $Fu{n}_{QCoh(X)}(QCoh(\ast ),QCoh(\ast ))\simeq QCoh(\Omega X)$ Proof $R$-linear Eilenberg-Watts.

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Remark An informal way to think about this functor is as as a Fourier-Mukai transform. Taking

• Input: $F$ - A $QCoh(X)$-linear endofunctor of $QCoh(\ast )$
• Output: ${\Phi }_F$ the Fourier-Mukai integral kernel of F, which lives on the stack $\Omega X$

Now, we observe that

• LHS has the structure of a $E_1$-monoidal, $QCoh(X)$-linear $\infty$-category, we’ll denote this by ${\Theta }_{(X,x)}$
• RHS has the structure of being the space of arrows in a cogroupoid object - concretely, it gives a cosimplicial category ${\mathcal{E}}_{(X,x)}^{\bullet }\in Fun(\Delta ,2QCoh(\ast ))$.

To be more specific, we observe that

• LHS controls 2-categorical codescent along $x$: ${\Theta }_{(X,x)}$ is the monad associated $2x^{\ast }⊢2x_{\ast }$.
• RHS knows about 1-categorical descent: $Tot({\mathcal{E}}_{X,x}^{\bullet })\simeq Rep(au{t}_X(x))$.

Putting this all together, we arrive at a conceptual picture:

• $(n+1)$-categorical codescent data can be described by their Fourier-Mukai kernels, which are $n$-categorical descent data

Next:

• explain how this gives rise to a notion of completion, possibly rephrased in Defn - Tannakian completions.
• do explicit example of doing reconstructions along $0:\ast \to {\mathbb{A}}^1$. 1-categorically this has to do with inverting/completing automorphisms, 2-categorically it has to do with … recollement?
• Taking ${\mathbb{G}}_m$-equivariant version of above, this says there’s some graded refinement of operations like inverting an endofunctor, and that this could have to do with internally filtered categories.