We consider Taylor approximation for functors from the small category of finite pointed sets $\Gamma$ to modules and give an explicit description for the homology of the layers of the Taylor tower. These layers are shown to be fibrant objects in a suitable closed model category structure. Explicit calculations are presented in characteristic zero including an application to higher order Hochschild homology. A spectral sequence for the homology of the homotopy fibres of this approximation is provided.

The paper studies the structure of functors $I\otimes F$ in the category of functors from finite dimensional ${𝔽}_2$-vector spaces to ${𝔽}_2$-vector spaces, where $F$ is a finite functor and $I$ is the injective functor $V\mapsto {𝔽}_2^{V^{*}}$. A detection theorem is proved for sub-functors of such functors, which is the basis of the proof that the functors $I\otimes F$ are artinian of type one.

We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical $\mathrm{Ext}$-computations as well as new results. In particular, we get a cohomological version of the “fundamental theorems” from classical invariant theory for $G{L}_n$ for $n$ big enough (and we give a conjecture for smaller values of $n$). We also study the “twisting spectral sequence” $E^{s,t}(F,G,r)$ converging to the extension groups ${\mathrm{Ext}}_{{𝒫}_{𝕜}}^{*}(F^{\left(r\right)},G^{\left(r\right)})$ between the...