By W. W. Sawyer
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Extra resources for A Concrete Approach to Abstract Algebra
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.
A Concrete Approach to Abstract Algebra by W. W. Sawyer