On the Andre-Quillen homology of Tambara functors

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 $\underset{‾}{R}$, and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over $\underset{‾}{R}$. 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 $\underset{‾}{R}$ are the same as maps out of the K"ahler differentials.