Defn - Tannakian completions

Definition

Let $X\stackrel{f}{\to }Y$ be a morphism in $Fun(CAlg,\mathcal{S})$, we’ll define $n$-Tannakian completion of $f$ to be the presheaf

$\widehat{nTn{k}_f}:=Fu{n}^{\otimes }(nDes{c}_f,nCa{t}_{(-)}{)}^{\simeq }\in Fun(CAlg,\mathcal{S})$ where $nDes{c}_f$ is the descent category for

$f^{\ast }:nQCoh(Y)\to nQCoh(X)$ That is,

$$nDes{c}_f:=coLMo{d}_Q(nCa{t}_B)$$

where $Q$ is the descent comonad.

Actions

• How to formulate a condition like $f^{\ast }\simeq (n-1)QCoh(X){\otimes }_{(n-1)QCoh(Y)}(-)$ is it some weaker variant of finiteness?

Basic structures

• The map $nDes{c}_f\to nQCoh(X)$ that coforgets comodule structure determines a map $i:X\to \widehat{nTn{k}_f}$
• Restricting along the map that computes descent data $\delta :nQCoh(Y)\to nDes{c}_f$ gives a map $j:\widehat{nTn{k}_f}\to nTn{k}_Y$ where the RHS is as the n-Tannakianization of $Y$, as in Defn- Tannakianization
• The

Action

• Is it the case that, there’s a commutative diagram:

\begin{CD} X @>{f}>> Y;\\ @ViVV @V\theta_YVV \\ \widehat{nTnk_f} @>{j}>> nTnk_Y; \end{CD}$$