Stefano Ambra (Università degli Studi di Milano) will give a talk on
The direction functor(s) for Schreier extensions of monoids
Abstract:
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 Gp/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 Gp/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-semimodules, 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 Gp/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.