By Hans Joachim Baues
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We now consider the category Top with the structure Top = (Top, cof, fib, I, P, 0, e). Here cof is the class of closed cofibrations in Top. 4) we see that Top satisfies (IP1) and (IP2). Strom proved that Top satisfies (IP3). 3) Proposition. Top is an IP-category. Moreover, Strom proved that Top satisfies (M2). 4) Corollary. 10). This example shows that it is convenient to consider the class cof of cofibrations as part of the structure of an 1-category. 1). 5) Proposition. Let u:C -* D be a map in C.
3) Definition. Let C be a category and let u: C -> D be a fixed map in C. We define the category C(u) = C(C -+ D) of objects under C and over D. Objects are triples (X, z, z) where u = U. x)-*(Y,y,f)isamap f:X -* Ysuchthat fz=y, yf=z. II C(u) has the initial object C C -> D and the final object C -> D -4 D. Pull backs and push outs in C(u) are given by pull backs and push outs in C. We define a natural cylinder I and a natural path space P which are functors 1, P:Top(u) -* Top(u). pp, u). We define W = P(X, x, z) by the pull back diagram in Top C w W XI pull w D where w = (u, ii).
3 Categories with a natural cylinder We here combine the notion of a cofibration category with the concept of abstract homotopy theory as introduced by Kan (1955). This concept relies on a natural cylinder object, see also Kamps. Under fairly weak assumptions on a natural cylinder we can derive the structure of a cofibration category. 1) Definition. An I-category is a category C with the structure (C, cof, 1, gyp). Here cof is a class of morphisms in C, called cofibrations. I is a functor C -> C together with natural transformations io, it and p.
Algebraic Homotopy by Hans Joachim Baues