alternative lecture notes - Rational points and algebraic cycles

A f → X c → X I → X × X
×{0}
A
c◦f
X I
(s,t)
A ×{1} A × I X × X
A
Lemma 8.3. Let A and X be both simultaneously fibrant and cofibrant. Then a map f : A →
B is in W if and only if it has a homotopy inverse, i.e., there exists g : B → A such that
f ◦ g ∼ id X and g ◦ f ∼ id A .
Proof. Skipped.
Using the lemma, we can prove the following important result:
Theorem 8.4. Let A and X be in C, and let A c be a cofibrant replacement for A. Let X f
be a fibrant replacement for X. Then
Hom C (A c , X f )
≃ Hom Ho(C,W ) (A, X).
∼
Example 8.5. Let A and B be simplicial sets. Every object is cofibrant. Fibrant objects
are Kan, so we take the Kan replacement.
f
X
9. July 30 (Schlank)
We now give two model categories of bounded complexes.
Example 9.1. In Ch ≥0
R-mod
(the category of chain complexes of R-modules in nonnegative
degrees), being cofibrant is being levelwise projective, and all objects are fibrant.
Example 9.2. In Ch ≤0
R-mod
(the category of chain complexes of R-modules in nonpositive
degrees), being fibrant is being levelwise injective, and all objects are cofibrant.
Let A be an R-module. Define K(A, n) ∈ Ch ≥0
R-mod
{
by
A if m = n,
(K(A, n)) m =
0 if m ≠ n.
Then
Hom Ch
≥0 (K(A, n) cof , K(B, m))/ ∼= Ext m−n
R (A, B).
R-mod
Let D be a small category, and let M be a model category. Objective: Define a model
category on M D .
Let G, F : D → M be two functors. A natural transformation h: G → F is a morphism in
M D . Say that h is a levelwise weak equivalence (or just a weak equivalence) if for all d ∈ D,
the morphism h d : G(d) → F (d) is a weak equivalence. Define levelwise fibration and levelwise
cofibration similarly.
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