isCohomological -- checks if a CpMackeyFunctor is cohomological

A Mackey functor $M$ defined over $C_p$ is said to be cohomological if the composite of restriction followed by transfer is identical to multiplication by $p$ as a map from the fixed module of $M$ to itself. We consider the category of cohomological Mackey functors to be the full subcategory on cohomological Mackey functors. An important result is the following:

Theorem: [TW95, 16.3] Every cohomological Mackey functor is a module over the fixed point Mackey functor $\underline{\ZZ}$, and every homomorphism between cohomological Mackey functors is a $\underline{\ZZ}$-module homomorphism.

Another way to say this is that the category of cohomological Mackey functors is the abelian subcategory $\text{Mod}_{\underline{\ZZ}}\subseteq \text{Mack}_{C_p}$. Carrying out computations (for instance Ext and Tor) internal to the abelian subcategory $\text{Mod}_{\underline{\ZZ}}$ leads to what are called cohomological Ext and cohomological Tor.

References:

• [TW95] J. Thevenaz & P. Webb, The structure of Mackey functors Trans. Amer. Math. Soc., 1995.