Cohomology, homotopy and n-categories

[pozorvlak]

Probably only michiexile and half_of_monty will be interested by this, but what the hell...

I thought that the totally awesome John Baez (quantum gravity bloke and cousin of Joan) had answered my prayers, and written an introduction to cohomology from the perspective of n-categories. But he's actually done something more interesting than that - I'm not exactly sure what yet, but it involves (-1)-categories and (-2)-categories, and relates a big wodge of homotopy theory to the idea of essential surjectivity in different dimensions. It's also pretty readable. Highly recommended, and I'd love to hear what some actual homo(log|top)ists think!

Comments

[(Anonymous)]

I probably need to get back to Jena before I can actually sit down at all with that paper (though, then again, possibly not - the Morse theory stuff next week seems to be slightly lighter than this wekeend is in program), but I've been almost seduced by the look of it already...

By the way - do you have a good three-sentence description of what a -1-category is, and why I should care?

[michiexile]

Sorry. Got logged out of LJ. This was me.

Three sentences?

[pozorvlak.livejournal.com]

OK, here goes...

Extending the idea that an n-category has (n-1) categories as hom-sets means that a 0-category (ie, a set) has (-1)-categories as hom-sets (1). But elements of a set are either equal (so there's an identity arrow) or not equal (so there's no arrow at all), so a -1-category is a truth value (2). This idea makes the periodic table conceptually cleaner; more interestingly, it allows us to construct Postnikov towers for n-groupoids just as we do for spaces, and a lot of the homotopy theory carries over (3).

Something like that, anyway... is it cheating to use semicolons?

I've just got to another exciting bit, where he uses group cohomology to classify certain n-groupoids: the classification of central extensions using H^2 then becomes the degenerate 1-dimensional case! This is probably less surprising to those well-versed in homological algebra, but I thought it was pretty cool.