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By Brown R., Higgins P., Sivera R.

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E. not just a subset but also a subgroup of P × P. An elementary result in group theory is that a congruence on a group P is determined completely by a normal subgroup of P. The corresponding result for groupoids is that a groupoid with a group structure is equivalent to a crossed module M → P where P is the group of objects of the groupoid. This family of equivalent structures – crossed modules, cat1 -groups, group objects in groupoids – gives added power to each of these structures. In fact in Chapter 6 we will use crucially another related structure, that of double groupoids with connection.

4] 11 There are two important, related and relevant differences between groupoids and groups. One is that groupoids have a partial multiplication, and the other is that the condition for two elements to be composable is a geometric one (namely the end point of one is the starting point of the other). This partial multiplication allows for groupoids to be thought of as “groups with many identities”. e. sets with a chosen base point. It is clear that graphs are more interesting than sets, and can reflect more geometry.

5 • • • • • • • • ...... 5: Extending to the lateral faces we get at the top face a map that looks like •dd • dd ~~ ~ dd ~ d ~~ ◦ ◦~ τlm hlm ◦ ◦d ~~ σlm ddd ~ ~ dd ~~ elm d ~ • • Thus, in particular, it is a map into Ui sending all vertices in A. 6] 19 If we do the above construction in each square of the subdivision, we get a top face of the cube that is an homotopy rel end points between two paths in the same classes as α and α ′ , and subdivided in such a way that each subsquare goes into some Ui sending all vertices into A.

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Nonabelian algebraic topology by Brown R., Higgins P., Sivera R.

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