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.

We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra....

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...