Defn- Tannakianization

Defn/Contruction Let $X\in Fun(CAlg,\mathcal{S})$, define

$nTn{k}_X:=nFu{n}^{\otimes }(nQCoh(X),nCa{t}_{(-)}{)}^{\simeq }\in Fun(CAlg,\mathcal{S})$

Comparison map and n-affineness

The identity functor $nQCoh(X)\to nQCoh(X)$ classifies an $X$-point of $nTn{k}_X$, that is, a map of stacks

${\theta }_X:X\to nTn{k}_X$ In components, the map $X(R)\to nTn{k}_X(R)$ sends

$\left(Spec(R)\stackrel{f}{\to }X\right)\mapsto \left(nQCoh(X){\stackrel{f}{\to }}^{\ast }nCa{t}_R\right)$ If the above map is an equivalence, $X$ is said to be $n$-affine. The map ${\theta }_X$ is said to be the n-affininization of $X$.

Non-commutative Generalizations

• if $A$ is $E_n$, above definition works verbatim to define $k$-Tannakianizations for $k\le n$ (where symmetric-monoidal functors are replaced by $(n-k)$-monoidal functors)