On the André–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 Kähler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor R_, and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over 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 Kähler differentials which satisfy the classical relation that derivations out of R_ are the same as maps out of the Kähler differentials.