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A point of information. These constructions are due to Charles Wells in this particular setting and to Jonathan Leech, (H-coextensions of monoids, vol. 1, Mem. Amer. Math. Soc, no. 157, American Mathematical Society, 1975) in the single object case, and McLane introduces the category of factorisations I think. Charlie Wells even pushes things a bit further than Baues. Hans does not seem to have known of that work. (Charles Wells, Extension theories for categories (preliminary report), (available from http://www.cwru.edu/artsci/math/wells/pub/pdf/catext.pdf), 1979. ) I have been meaning to have a go at this entry as I have written up a modern version of Wells especially in the non-Abelian case. There is a very nice interpretation of Natural System as a lax functor. (I will do this some time…. but I can make the notes available to anyone interested.)
These constructions […] in this particular setting […]
Are you referring to the nLab entry category of factorizations?
Yes. Baues and Wirshing defined this but Wells had followed Leech in using it to define a cohomology of categories/monoids. That definition relates to properties in monoid theory as such. Wells also made comments about non-Abelian cohomology in this context.
A natural system of Abelian groups corresponds to an Abelian group object in the category of O-categories over the Base (with functors fixed on objects). Wells had noted, more or less, that group objects in that category gave something lax. In fact I fiddled about last year and showed that the ’correct’ analogue of a natural system in a non-Abelian context would be a lax functor from the base $B$ to $Gp[1]$, the monoidal category suspension of the ‘Cartesian monoidal’ category of groups. The proof is easy once the concepts are sorted out and is the Grothendieck construction (surprise, surprise) but in disguise. These then correspond to group objects in $Cat_O/B$ where $O$ is the set of objects of $B$ and $Cat_o$ is the cat of small ’O-objected’ categories etc. Wells cohomology then classifies torsors for the given coefficient group. (That gives the idea. The proofs are fairly easy although I took a long time to find some of them!) I have not taken the ideas further than Wells did yet.
I have changed a few things in the entry. The main facts were already in Baues-Wirsching cohomology.
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