From: miguel@math-cl-n03.math.ucr.edu (Miguel Carrion) Subject: Re: Cech cohomology and bundles Date: Mon, 10 Dec 2001 03:57:36 +0000 (UTC) Newsgroups: sci.physics.research Summary: Morphisms in 2-categories In article <9uf1f3$1l2$1@glue.ucr.edu>, Toby Bartels wrote: >Miguel Carrion wrote in part: > >>Toby Bartels wrote: > >>I have mended my evil ways. > >You sure have! You seem to have come around to almost all of my notation. >How unlike the evil jim dolan you are ^_^! Yes, I don't know more category theory than you do, so I yield in things like the order of composition. I am less likely to yield on "important" stuff :o) >It probably does, but if so, then I charge that >it *doesn't* make sense -- at least not in a fully taoist way -- >to allow g_ii(x) to be only idempotent in such cases. I guess I'll have to accept "the Tao of mathematics" as a method of proof :-) > >>Now I think we have exhausted 1-bundles, so I am ready to talk about >>2-bundles with a structure 2-category. But this post is already too >>long, so I'll put them in my next post. > >Well, I still have to do the cohomology, but you don't. > I don't know what you mean by that, but maybe you'll elaborate... Anyway, time to crank up and do 2-bundles with a 2-category fibre. What is a 2-bundle? As before, we let {M_i} be a locally finite cover of M by contractible open sets with contractible intersections, and let M_ij...z denote the intersection of M_i, M_k, ..., M_z. Now, consider the 2-category where objects are points x_i in M_i, morphisms are paths gamma_ij : x_i -> y_j in M_ij and 2-morphisms are "triangles" h_ijk : alpha_ij beta_jk => gamma_ik in M_ijk and such that alpha_ij : x_i -> y_j, beta_jk : y_j -> z_k and gamma_ik : x_i -> z_k. These 2-morphisms satisfy a compatibility condition modelled after a tetrahedron: w_i w_i /|\ / \ / | \ / \ v | v v ^ v / | \ / // \ / | \ / \ x_j => v ^ z_l = x_j --->--- z_l \ |// / \ / \ | / \ /\ / v | ^ v || ^ \ | / \ / \|/ \ / y_k y_k h_ijk h_ikl = h_jkl h_ijl If we write this out in full, we get alpha_ij beta_jk gamma_kl h_ijk w_i ---->---- x_j ---->---- y_k ---->---- z_l ==>== delta_ik gamma_jk h_ikl eta_il w_i ---->---- y_k ---->---- z_l ==>== w_i ---->---- z_l and, for the r.h.s., alpha_ij beta_jk gamma_kl h_jkl w_i ---->---- x_j ---->---- y_k ---->---- z_l ==>== alpha_ij zeta_jl h_ijl eta_il w_i ---->---- x_j ---->---- z_l ==>== w_i ---->---- z_l This is just the associative property for 2-morphisms. Special cases of 1-morphisms include the constant path x_ij : x_i -> x_j and paths gamma_ii : x_i -> y_i. Any path between x_i and y_j where M_ij is empty can be written as a finite composition of maps f the form gamma_ij. (A question: is now the time to say that this composition is unique up to an invertible 2-morphism?) Special cases of 2-morphisms include the case where y_j is an intermediate point of gamma_ik : x_i -> z_k, and gamma_ij : x_i -> y_j and gamma_jk : y_j -> z_k are subpaths of gamma. When we had a 1-bundle and a category of points and paths, we were forced to say gamma_ik = alpha_ij beta_jk (Toby's "fundamental commutative triangle"), but now it seems that the right thing to do is to ay it is no longer commutative, but to introduce a special 2-morphism h_ijk : gamma_ij gamma_jk => gamma_ik called "subdivision of paths". John mentioned this morphism as a morphism in the category of finite embedded graphs during his last seminar. Special cases of this are when y = z and gamma_jk = y_jk is the constant path: we have h_ijk : gamma_ij y_jk => gamma_ik and h_ijk : x_ij gamma_jk => gamma_ik. An even more degenerate case is h_ijk : x_ij x_jk => x_ik. We also have the case of a non-degenerate triangle which is contained in a double or single intersection: h_ijj : alpha_ij beta_jj => gamma_ij where alpha : x -> y and beta : y -> z, and so on (there are too many of these to list). Finally, maybe M_ijk is empty, so h_ijk must be constructed by covering the triangle with lots of open sets and composing the elementary triangles that arise. I guess the associative property / tetrahedron identity guarantees that this is independent of how h_ijk is chopped up into bits, but I'll let Toby reassure me of that. There are some issues having to do with nontrivial homology, but this is already too long, and I haven't even defined a connection. I'll do that in the next post. Regards, Miguel -- }--------------------------------------------------------------------- | homepage: | International Association of Physics Students, IAPS | url: e-mail: info@iaphys.org