I wish to understand if is possible to build a function 'bridge' between $μ$ measure and $act$ action definitions because I did not find any direct connection or a way to build a own category.
Only universal connection that I find is only for finite dimensional Lie group $G$. Ehresmann or Cartan connection is still no sufficient because it works like a version of notion of connection.
Concept of a working $μ$-$act$ category is hard to figure out because I really don't understand from where point we deep start within: group cohomology, Homotopy category of chain complexes, inhabited set ?
An inhabited set is the special case of an internally inhabited object in the topos $\text{Set}$
Please guide me to formalize the idea by identifying the most correct categories.