Saturday, 1 August 2009

higher category theory - 2-groups are to crossed modules as 2-categories are to...?

Given a 2-group mathcalG, you can construct a crossed module (G,H,t,alpha) and vice versa.



Is there something similar you can say for strict 2-categories?



In a personal attempt to understand strict 2-categories, I ended up constructing a speculative conceptual tool (whose validity remains to be seen) that I call the boundary of a 2-morphism. I've written up some raw notes here:



The basic idea is that given morphisms f,g:xtoy and a 2-morphism alpha:fRightarrowg, we define its boundary as an endomorphism



partialalpha:ytoy



satisfying



partialalphacircf=g.



When the source of the 2-morphism is an identity morphism, then we have



partialalpha=t(alpha),



which seems to relate things well to cross modules when all morphisms are invertible.



I'm curious if there is anything like a crossed module, but where we're not dealing with groups and morphisms are not invertible. What I'm trying to cook up seems like it might be related to such a thing if it exists.



Any thoughts and/or any comments on my notes would be greatly appreciated.



PS: Apologies in advance if my writing is not very clear. I'm not a mathematician, but am trying to teach myself some basic higher category theory.

No comments:

Post a Comment