# On the Andre-Quillen homology of Tambara functors

## TL;DR

This paper extends classical algebraic concepts like derivations and K"ahler differentials to the setting of Tambara functors, establishing foundational structures and equivalences in equivariant algebra.

## Contribution

It introduces Mackey functor objects in Tambara functors, generalizes derivations, and defines K"ahler differentials within this equivariant framework.

## Key findings

- Equivalence between Mackey functor objects and modules over Tambara functors.
- Generalization of derivations to Tambara functors.
- Definition of K"ahler differentials satisfying classical relations.

## Abstract

We lift to equivariant algebra three closely related classical algebraic concepts: abelian group objects in augmented commutative algebras, derivations, and K\"ahler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor $\underline{R}$, and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over $\underline{R}$. We then describe the natural generalization to Tambara functors of a derivation, building on the intuition that a Tambara functor has products twisted by arbitrary finite $G$-sets, and we connect this to square-zero extensions in the expected way. Finally, we show that there is an appropriate form of K\"ahler differentials which satisfy the classical relation that derivations out of $\underline{R}$ are the same as maps out of the K\"ahler differentials.