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By Hans Joachim Baues

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This is the initial object * in the category of graded algebras since we always have is*-+A, i(1)= 1. 2) Definition. An augmentation of an algebra A is a map &:A -* * between graded algebras. Let A = kernel (s) be the augmentation ideal. The quotient module QA = is the module of indecomposables. Here denotes µ(A (D A). An augmentation preserving map f between algebras induces Qf :QA- QB. 3) Definition. For a positive graded module V we have the tensor algebra T(V) _ (@ V®n, n20 where V on = VQx .

3). 2) Theorem. 7), is a fibration category in which all objects are fibrant and cofibrant. Proof. 4). 2) are strictly dual. 3) cof = maps in Top* which are cofibrations in Top, fib = maps in Top* which are fibrations in Top, we = maps in Top* which are homotopy equivalences in Top. 3). 4) Theorem. 3) is a cofibration category in which all objects are fibrant models. The well pointed spaces are the cofibrant objects. Proof. 2). 5) Theorem. 3) is a fibration category in which all objects are fibrant.

We obtain path objects and the notion of homotopy in a fibration category as follows: By (F2) there exist pull backs AxBY )A Y f B in F. 3) (lA, lA):A-+A X BA the diagonal map which is dual to the folding map. 4) A +P-A xBA 1 q 12 I Axioms and examples B. 5) H\ / (a, II) X We call H a homotopy from a to fl over B. Here we assume that X is a cofibrant object in F. 5). 9), and with homotopy equivalences as weak equivalences is a fibration category. This is proved in §4 below. Our definition of a cofibration category (resp.

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