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IB 1 '0 l G I Axioms and examples 22 we obtain the extension E:IA -+ U by (13). Then we set G# {v} = {Eil I. Here G# is well defined, since for a homotopy H: v a v' under B we have the commutative diagram (by (12)) AuIBuA - I (AuIBuA) 10 (E,GIp,E') i IA . 1 H U We choose a homotopy extension E. Then Ei1:Ei1 ^-, E'i1 under B.

1) Definition. A chain complex V is a graded module V with a map d: V -+ V 40 I Axioms and examples of degree - 1 satisfying dd = 0. A chain map f : V -* V between chain complexes is a map of degree 0 with df = fd. Let ChainR be the category of chain complexes and of chain maps. The homology HV is the graded module HV = kernel d/image d. The homology is a functor from ChainR to the category of graded modules. We may consider a graded module as being a chain complex with trivial differential d = 0.

Then we see that Ei 1 = A has the properties in (16). 5). 13) Lemma. p:IBA-+ A is a homotopy equivalence. Proof. We have the push out diagram A Since p is a homotopy equivalence also p is one by (C2). § 3a Appendix: categories with a natural path object The dual of an 1-category is a P-category (C, fib, P, e), where fib is the class of fibrations, P is the natural path object and e is the final object in C. We write I Axioms and examples 28 PX = X' for an object X in C. 2). , (P5) which are dual to the axioms (I1), ...

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Algebraic Homotopy by Hans Joachim Baues

by Ronald

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