formal codeformations as mapping spaces

Idea

Where we’re going: formally completing co-deformations as a local system

We’ll consider a notion of deforming $M$ - a left $B$-module, along $f:A\to B$ -a map of $E_1$ rings.

$$coDef(M,f)\to \widehat{coDef(M,f)}$$

Where

• LHS is the category of codeformation Defn-codeformation.
• RHS the an $\infty$-groupoid(!) that we’ll define as

$$\widehat{coDef(M,f)}:=Ma{p}_{Al{g}^{(1)}}(En{d}_A(B),En{d}_B(M))$$

We show below that this admit a right-adjoint, which in nice cases will turn out to be fully faithful.

Remarks

• This map is a local system. We can do things like calculate its homology, cohomology, make use of $\infty$-semiaddivity in monochromatic settings, etc.
• The projection map in the local system implements the operation of “formally completing”. That is, it sends a codeformation to a “formal codeformation”.
• The total space of this local system is a “simple kind of category”: it the sort of thing that arises as the fiber at a single object of a right-adjoint functor between stable $\infty$-categories.

Relationship to moduli of objects We’ll denote $\mathbb{D}(B/A):=En{d}_A(B)$, to suggest that it should be thought of as an unpointed $E_1$-Koszul duality. From this perspective, the right adjoint to the above provides a map - which will turn out to be fully faithful when $f$ is sufficiently nice ($B$ is nice relative to $A$).

$\{\text{Actions of D(B/A) on M}\}\to \{\text{codeformations of M along A→B}\}$ This is a generalization of the type of results proved in DAG-X about deformations of objects - where similar outcome was obtained by thinking explicitly about deforming objects in a $k$-linear setting: however , those results are about deformations.

What does the completion map do?

The data of a morphism of algebras

$En{d}_A(B)\to En{d}_B(M)$ is the data of an action of $En{d}_A(B)$ on $M$.

Now $En{d}_A(B)$ represents the monad for the adjunction

${\int }^{(-)}B:LMo{d}_B\leftrightarrow LMo{d}_A^{op}:Ma{p}_A(-,B)$ Hence, the data of an algebra map above is precisely co-descent data along this adjunction. Our notion of “completing a codeformation” has the following heuristic:

• To complete a codeformation $(E,\theta )$ means to compute its codescent data
• To embed formal deformations back into deformations, compute the object glued out of codescent data.

Construction

Descent along the adjunction ${\int }^{(-)}B:LMo{d}_B\leftrightarrow LMo{d}_A^{op}:Ma{p}_A(-,B)$ gives comparison map

$$Ma{p}_A(-,B):LMo{d}_A^{op}\stackrel{\widetilde{Ma{p}_A(-,B)}}{\to }LMo{d}_{En{d}_A(B)}(LMo{d}_B)\stackrel{forget}{\to }LMo{d}_B$$

Taking the fiber of this along $\ast \stackrel{M}{\to }LMo{d}_B$ gives the desired comparison map.

LMod_{End_A(B)}(LMod_B) \times_{LMod_B}\{M\} \simeq Map_{Alg^{(1)}}(End_{A}(B), End_{B}(M))$$ **Actions** - this notion categorifies, following how descent categorifies. - This suggests a dual notion of a formal Deformation (more familiar geometrically, but a less intuitive structure): a formal deformation is a lift to a comodule over descent comonad. (Because not all objects have coendomorphism coalgebras, this is not merely a map of co-algebras, hence a bit harder than before).