defn - functorial algebra tower

Definition: A functorial tower of algebras $\rho$ is the data of

• a functor

$${\rho }_{\bullet }:CAl{g}_{nu}\to Fun({\mathbb{Z}}^{\ge },CAl{g}_{nu})$$

• a natural transformation $T_{\rho }:\Delta \to {\rho }_{\bullet }$

i.e. it’s the data for each $I\in CAl{g}_{nu}$ a tower of non-unital $E_{\infty }$ algebras under $I$, functorial in $I$.

Examples include $E_n$ Goodwillie towers for all $n$, the Cech/Amitsur filtration, and our conjectured $E_n$ Amitur filtrations.

Definition: An algebra $I\in CAl{g}_{nu}$ is said to be $\rho$-complete if the map

$$I\to \underset{n}{\lim }{\rho }_n(I):={\rho }_{\infty }(I)$$

is an equivalence.

Generalizations: This obviously generalizes with $CAl{g}_{nu}$ replaced by an arbitrary $\infty$-category.