By Hans Joachim Baues
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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 . . Qx V is the n-fold product, V ®0 = R. We have inclusions and projections of graded modules V®n->T(V),V®n. in Pn The tensor algebra is an algebra with multiplication given by V®n Qx V®m = V®("').
Axiom (F4) is given as follows. (F4) Axiom on cofibrant models: For each object X in F there is a trivial fibration MX X where MX is cofibrant in F. Of course, a cofibration category has properties which are strictly dual to the properties of a fibration category and vice versa. It turns out that this is a good axiomatic background for many results which satisfy the EckmannHilton duality. Any result in a cofibration category which follows from the axioms (C 1), ... , (C4) corresponds to a dual result in a fibration category which follows precisely by dual arguments from the dual axioms (F 1), ...
5) Theorem. 3) is a fibration category in which all objects are fibrant. 2). Proof. 4). 4). 6) we have a different internal structure of Top* as a cofibration category and as a fibration category. Next we describe some examples for which the weak equivalences are not homotopy equivalences in Top. 12) we = weak homotopy equivalences = maps f :X -* Y which I Axioms and examples 36 induce isomorphisms f,: nk(X, x0) = 71k(1', J xo) on homotopy groups for k > 0, x0 eX. 6) Theorem. The category Top with the CW-structure (cof, we) is a cofibration category in which all objects are fibrant models.
Algebraic Homotopy by Hans Joachim Baues