Algebraic Homotopy by Hans Joachim Baues
By Hans Joachim Baues
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Extra info for Algebraic Homotopy
The empty space 0 is the initial object in Top and the point e is the final object in Top. Moreover, in Top exist push outs and pull backs. 2) Proposition. With the notation in § 0 the category Top has the structure of an I-category and of a P-category. Proof. Only (14) is not so obvious. We prove (I4). In a dual way we prove the P-category structure, see page 133 in Baues (1977). Let a:I x I I x I be a homeomorphism which is given on the boundary by the sketches: ,,1 _ V U w1 o c a ,U a b A w a b c One easily verifies that is B -+ A is a cofibration if for the push out diagram A--+ Au1B io )IA IB io with jo = (io, Ii) there is a retraction r with rjo = 1.
Moreover, if M is proper (M, cof, we) is a cofibration category and (M, fib, we) is a fibration category. Proof: (C3) follows directly from (M2). We now prove (C4). For any object X we have by (M2) a factorization X >-^--* R -> e of X -+ e. We claim that R is a fibrant model. In fact, for each trivial cofibration R >-=> Q we have by (M 1) 16 I Axioms and examples the commutative diagram Q` )e This shows that each e-fibrant object in M is a fibrant model. 1) is essentially consistent with Quillen's definition of fibrant objects.
2). 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.