Identify higher coherences in the punctured cube of twisted Gysin maps

Identify the higher coherences of the diagram of cohomology groups indexed by the poset of non-empty subsets I ⊂ {1,…,n}, whose one-morphisms are twisted Gysin maps E(∂_I X(#I)) → E(∂_J X(#J)) for inclusions I ⊂ J, arising in the computation of the weight filtration W_*E(X − ∂X) via the cofibre formula. Specifically, construct and describe the coherent higher morphisms that make this diagram into a functor given by coherent Gysin maps, thereby establishing a fully coherent functorial model for the punctured cube used in Equation (1).

Background

In the computation of the weight filtration for logarithmic cohomology of a complement of a relative snc divisor, the authors use a punctured cube indexed by non-empty subsets I of {1,…,n}, built from intersections ∂_I X and Gysin maps. While one-morphisms are available via twisted Gysin maps, the full coherent functorial structure requires identifying higher coherences.

The authors note that, although it is expected that this functor should be given by coherent Gysin maps, the higher coherences have not yet been identified, preventing a complete functorial description of the punctured cube in the filtered context.