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I have to prove that slice category $\mathcal{C}/X$ for $\mathcal{C}$ a Quillen model category is also a Quillen model category. I proved 2 out of 3 axiom, but I'm stuck with the retract axiom. For me, it's obvious that diagram commutes (morphism in slice category), because of definition of retract.

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    What are your proposed classes of weak equivalences, fibrations, and cofibrations?2012-10-17
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    W.e. is diagram that comutes $h_1=h_2f$ and f is w.e. , $f:A->B, h_1:A->X, h_2:B->X$.Analog for fibration and cofibration2012-10-17
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    You're making potential answerers work hard to understand your problem. That's suboptimal for getting responses. Anyway, do you mean $f$ is a weak equivalence between $A$ and $B$ in the base category $C$ (apparently a model category) we're taking slices over $X$ in?2012-10-17
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    I'm sorry because I'm not very clear in explaining,english is not my mother language, I'll try to do so now on. Yes,f is weak equivalence between A and B in the base C and slice category is $C/X$.2012-10-17
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    No worries. Next time, maybe it would help to write your question out in your native language, and decide whether it includes all the necessary information before translating into English. I've answered below.2012-10-18

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