alternative lecture notes - Rational points and algebraic cycles

It is common when describing a model category to specify W and only one of FIB and
COF — the lemma shows that the other is determined.
Let A ∈ C. Let f : A ∐ A → A. It factors as A ∐ A ↩→ A × I ∼ ↠ A. (This is the definition
of A × I, which is a single symbol; there is no I.) This A × I is called the universal cylinder
object of A.
More generally, given a factorization A ∐ A → A ∧ I ∼ → A, call the object A ∧ I a cylinder
object.
Given two maps f, g : A → B, we will say that f is strictly left homotopic to g by A ∧ I
and write f l ∼ g if there exists H completing the diagram below:
A ∐ f ∐ g
A B
H
A ∧ I.
The dual definition: Let X ∈ C. Let ∆: X → X × X. We have X ∼
↩→ X I ↠ X × X and
X I is called the universal path object. Given any decomposition X ∼ → X ∧I → X × X, call
X ∧I a path object.
Given f, g : Y → X, write f r ∼ g (right homotopic) if there exists H completing the diagram
below:
Y
X ∧I
H
X × X
Proposition 8.2. Let A ∈ C be cofibrant, let X ∈ C be fibrant, and let f, g : A → X. Then
(1) If f l ∼ g by some A ∧ I, then f l ∼ g by A × I.
(2) l ∼ is an equivalence relation on maps A → X
(3) f l ∼ g if and only if f r ∼ g.
Proof.
(1) Given A ∐ A → A ∧ I ∼ → A, factor the second map as A ∧ I ∼
↩→ A ∧ I ′ ↠ A; then the
last morphism must also be in W .
Then
A ∐ A
A ∧ I
∼
A ∧ I ′
H
f ∐ g
X
∗
A ∐ A A ∧ I
′
∼
∼
A × I A
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