defn-Koszul-formal neighborhoods

We’d like to define a functor

$$Nbh{d}_{Spec(R)}:CAlg\to S$$

That captures the idea $Nbh{d}_{Spec(R)}(A)\simeq \left[\frac{Map(R,A)}{\text{infinitesmal equivalences}}\right]$ Here’s one approach.

Fix $\rho$ be a functorial algebra tower, as in defn - functorial algebra tower.

Defn: An infinitesmal equivalence between $R\to A$ and $T\to A$ is data of an equivalence of towers

${\rho }_{\bullet }(I_R)\to {\rho }_{\bullet }(I_T)$ We should try to write this as a groupoid quotient, think about composition structure of these. The order-by-order structure, when using Goodwillie calculus, is reminescent of PD-structures.

Idea: can we phrase this in terms of the groups $aut(A/R)$? (Use a $E_1$ Goodwilllie tower on these groups?), also as in constr - unpointed toposic Koszul duality