On weak maps between 2-groups

Details On weak maps between 2-groups On weak maps between 2-groups

2005-06-15

arxiv.org/abs/math/0506313v3

Mathematics

Category Theory

A news section added and Section 9 expanded. Some minor errors corrected (and possibly new ones added). 48 pages

Scientific paper

We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way find a bicategory that is naturally biequivalent to the 2-category of pointed homotopy 2-types. We indicate how certain standard notions of 2-group theory (e.g., kernels, cokernels, extension of 2-groups, and so on) find a simple description in terms of butterflies. We also discuss braided and abelian butterflies.

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