By a classical result of Dominique Bourn, the second cohomology group of a group \( G \) with coefficients in a \( G \)-module \( A \) can be recovered by the fibre over \( A \) of the so-called direction functor \( d=d_0 \), when it is applied to the slice category \( G_p/G \). The higher dimensional group cohomology groups are similarly described by means of higher dimensional direction functors \( d_n \) (\( n>0 \)), introduced by Dominique Bourn and Diana Rodelo. In this talk, I would like to present a generalization of the above direction functor in the case of Schreier extensions of monoids, showing that we can consider a new functor \( D \) which coincides with \( d \) on the slice category \( G_p/G \), when \( D \) is applied to group extensions, and which retains the same good categorical properties. The fibres of \( D \), which are endowed with a canonical symmetric monoidal structure, now describe the second cohomology monoids of a monoid \( M \) with coefficients in an \( M \)-semimodule, as introduced by Alex Patchkoria, with applications (among others) in the arithmetic theory of the Brauer group and in Galois cohomology. Time permitting, I will also show that by considering an appropriate class of (Schreier) internal categories in the category of monoids, one can define higher dimensional direction functors \( D_n \) (generalizing the classical \( d_n \)'s on \( G_p/G \)), pointing towards a complete cohomology theory given entirely in terms of direction functors.

This is based on joint work with Andrea Montoli and Diana Rodelo.