2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/164695We 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.A news section added and Section 9 expanded. Some minor errors corrected (and possibly new ones added). 48 pagesCategory TheoryGeometric TopologyQuantum Algebratext