Proposition 5.96.
The category \(\Exc^1(\An^{\fin},T)\) is equivalent to the total category of \(T\)-parametrized spectral torsors:
\[\{(X,E,s) \mid X \in T, E \in \Sp(T_{/X}), s \in \Gamma(X,E)\}.\]
Here the right-hand side denotes the category whose morphisms consist of a map of base objects together with a compatible map of the pulled-back spectra carrying one section to the other.
Proof idea
Adjoining a disjoint basepoint gives a functor \(\An^{\fin}\to\An_*^{\fin}\). Under the preceding proposition, the additional datum required to descend a pointed excisive functor along this construction is a section of the underlying parametrized spectrum. This identifies unpointed \(1\)-excisive functors with parametrized spectral torsors.