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We include the proof since it is a good example of the axioms at play. 3 Let M be a model category. If f : A → B is a map between cofibrant objects, then there exists a (functorial) diagram A GB }b b y } } }}∼ = }}jf Cf o ∼ o B f d2 d i dd f dd d2 pf where if is a cofibration, pf is a trivial fibration and jf a trivial cofibration. Dually, if f : A ← B is a map between fibrant objects, then there exists a (functorial) diagram A –do – dd ddPf dd f B ~~ ~ ~ ~~∼ = ~ ~Qf ~  Zf ∼ G G B If where If is a trivial cofibration, Pf is a fibration and Qf a trivial fibration.

4]). 4 Prove that if Y ∈ T op, then sing Y is fibrant. 5 Note that a trivial fibration actually is a fibration. 6 One may show that being a trivial fibration is equivalent to being both a fibration and a weak equivalence. 7 Our definition of fibrations is equivalent to saying that a fibration is a map which has the right lifting property with respect to injections that are weak equivalences. To see this one has to show that inclusions that are weak equivalences can be built out of the filling of horns (more precisely, they are retracts of injections X(0) → X(i) where each X(i − 1) → X(i) are pushouts of disjoint unions of Λk [n] ⊆ ∆[n]’s).

I. 5, just that the morphism objects stays within MS (they are motivic spaces, not just spaces). We say that MS is a monoidal model category. However, the projective structure is definitely not the structure we are interested in on MS . Firstly we have to take into account some topology, and secondly we will want the affine line to be contractible. This can be fixed as follows. The ideology is that we specify the fibrant objects as those objects having some desired property – at least up to homotopy – and model weak equivalences and cofibrations on them.

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A Concrete Approach to Abstract Algebra by W. W. Sawyer

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