Proposition 5.95.
The category \(\Exc^1(\An_*^{\fin},T)\) is equivalent to the total category of \(T\)-parametrized spectra:
\[\int_{X\in T}\Sp(T_{/X}).\]
Proof idea
A \(1\)-excisive functor on finite pointed animae determines its base object \(X\) by evaluation at the zero object. Its reduced part is an excisive reduced functor in the slice \(T_{/X}\), hence a spectrum object of that slice. Conversely, a spectrum object in \(T_{/X}\) determines such a reduced excisive functor, and adjoining the base \(X\) recovers the original functor. These constructions are inverse and natural in \(X\). This is the internal version of Goodwillie's classification of \(1\)-excisive functors; see [Goodwillie 2003, Section 5] and [Anel et al. 2025, Section 4.3].
References
- Thomas\bibnamedelima G. Goodwillie. Calculus. III: Taylor series. Geom. Topol., 7, 645–711. 2003.
- Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Left-exact localizations of $\infty$-topoi III: The acyclic product. 2025.