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.
Three sentences?
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.