# New PDF release: Algebraic Geometry. Proc. conf. Chicago, 1980 By A. Libgober, P. Wagreich

ISBN-10: 3540108335

ISBN-13: 9783540108337

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There is also no benefit today in insisting that the group concept is more fundamental than that of groupoid; one uses each at the appropriate place. It is as well to distinguish the sociology of the use of a mathematical concept from the scientific consideration of its relevance to the progress of mathematics. It should also be said that the development of new concepts and language is a different activity from the successful employment of a range of known techniques to solve already formulated problems.

2, which implies C = P/µ(M). Since [m, n]p = [mp , np ] for m, n ∈ M, p ∈ P, we have [M, M] is P-invariant, so that P acts on Mab . However in this action µ(M) acts trivially since if m, n ∈ M then mµn = n−1 mn = m mod [M, M]. 2] 33 Thus for any crossed module (µ : M → P) with C = Cok µ, π = Ker µ we have an exact sequence of C-modules π −→ Mab −→ (µM)ab −→ 1. The first map is not injective in general. To see this, consider the crossed module χ : M → Aut(M) associated to a group M. Then π = Ker χ = ZM, the centre of M.

6) and then can be used to prove a 2-dimensional van Kampen Theorem. 8. 7] Nonabelian Algebraic Topology It is much more difficult to follow this route in the general case and a more roundabout method is developed in Chapter 14. The algebra to carry out this argument in dimension n is given in Chapter 13. It is interesting that such a complicated and subtle algebra seems to be needed to make it all work. We emphasise the the purely algebraic work of Chapter 13 is essential for the applications in the following two chapters, and so for the whole of Part II.