Motivic Homotopy Theory: Lectures at a Summer School in by Bjørn Ian Dundas (auth.), Bjørn Ian Dundas, Marc Levine,

By Bjørn Ian Dundas (auth.), Bjørn Ian Dundas, Marc Levine, Paul Arne Østvær, Oliver Röndigs, Vladimir Voevodsky (eds.)

Voevodsky is among the founders of the idea and bought the Fields medal for his paintings, and the opposite authors have all performed very important paintings within the subject.

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Additional info for Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002

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Continuous means that X induces a map of morphism spaces MS (v, w) → MS (X(v), X(w)) ∈ MS (not just on the underlying sets). Such functors are referred to by the category theorists as “enriched functors”. You have seen before how this is useful when we discussed ˜ ] → Z[X ˜ ∧ Y ]. This the Eilenberg-Mac Lane spectrum, and got a map X ∧ Z[Y ˜ – when considered as a functor on S∗ – induces maps is done by noting that Z on function spaces, so that you get a chain S∗ (X ∧ Y, X ∧ Y ) ∼ = S∗ (X, S∗ (Y, X ∧ Y )) ˜ ], Z[X ˜ ˜ ], Z[X ˜ ∧ Y ])) ∼ ∧ Y ]) .

Whose colimit (union) we call X∞ . We let X ιf G Zf φf GGY ∼ be X → X∞ → Y Note that X → X∞ is an inclusion. The important thing is that X∞ → Y is a trivial fibration. To see this, we need to observe that ∂∆[n] is small (much more about small objects later), which for our current purposes implies that a map ∂∆[n] → X∞ must actually factor through Xm ⊆ X∞ for some (possibly very big) integer m. So if we have a square of the sort ∂∆[n] G X∞ φf incl.  ∆[n]  GY and ask for a lifting, let m be such that ∂∆[n] → X∞ factors through Xm → X∞ and notice that by the very construction of Xm+1 we have a commutative diagram Algebraic Topology 33 G Xm .

X(i) ∈ S. Every k-simplex in A must necesLet f : A → lim→ − X(i) = sarily map to X(i)k for some i depending on the simplex, but since there are only finitely may non-degenerate simplices we may choose a specific i such that they all map to X(i). If x ∈ An is any simplex, there is a unique non-degenerate simplex y such that x = φ∗ y for some surjective φ ∈ ∆, and so f (x) = f (φ∗ y) = φ∗ f (y) must also be in X(i) (since X(i) ⊆ X(j) is a simplicial map). Remark 1. We have the small object argument for any A ∈ S, provided the cardinality of the indexing of the colimit is sufficiently big.

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