Defn-codeformation

Fixing:

• $f:A\to B$ a map of $E_1$ algebras
• $M\in LMo{d}_B$ left-representation of B

A co-deformation is a kind of a dual-notion to deformation of modules, where tensoring (Tor) is replaced by mapping spaces (Ext).

Defn (codeformations)

A codeformation of $M$ along $f$ is data

• $E\in LMo{d}_A$
• $\theta :Ma{p}_A(E,B)\stackrel{\sim }{\to }M$ an equivalence of left $B$-modules.

These assemble into an $\infty$-category

$coDef(M,f):=LMo{d}_A^{op}{\times }_{LMo{d}_B}\{M\}$ where the RHS denotes the fiber of the functor $Ma{p}_A(-,B)$ at $M$.