alternative lecture notes - Rational points and algebraic cycles

Let X and Y be two nice spaces. A weak equivalence is a map f : X → Y that induces
an isomorphism on π 0 , and on π n with respect to all x ∈ X (and f(x) ∈ Y ). This is not an
equivalence relation. Two spaces are weakly equivalent if they can be connected by a zigzag
of weak equivalences.
Definition 6.7. Let Sing : Top → S be the functor sending a topological space Z to
Hom Top (|∆ n |, Z).
Sing is right adjoint to the realization functor | |.
Claim 1: The counit map | Sing(Z)| → Z is a weak equivalence.
Claim 2: If X, Y ∈ S are such that |X|, |Y | are weakly equivalent, then |X|, |Y | are
homotopy equivalent.
Definition 6.8. A relative category is a category C together with a subcategory W ⊆ C
such that W contains all objects (and all identities).
Examples 6.9.
(1) C = Top and W is the subcategory of all homotopy equivalences
(2) C = Top and W is the subcategory of all weak equivalences
(3) C = S and W contains f : X → Y if |f| is a homotopy equivalence (or equivalently
by Claim 2, weak equivalence)
(4) C is the category of chain complexes of abelian groups, and W is the subcategory of
chain equivalences (there are variants: bounded below, bounded above, unbounded
on both sides)
(5) C is the category of chain complexes of abelian groups, and W is the subcategory of
quasi-isomorphisms
(6) C is the category of G-spaces and W contains f : X → Y if the underlying map of
spaces is a weak homotopy equivalence.
Say that (C, W ) is weakly homotopical if
(i) W contains all isomorphisms, and
7. July 23 (Schlank)
(ii) for every pair of composable maps X → f Y
is the third.
Examples 7.1.
g → Z, if two of f, g, g ◦ f are in W , then so
(1) C = Top and W is the subcategory of all homotopy equivalences
(2) C = S and W contains f : X → Y if |f| is a homotopy equivalence (or equivalently
by Claim 2, weak equivalence)
(3) C is the category of chain complexes of abelian groups, and W is the subcategory of
quasi-isomorphisms
(4) C is the category of G-spaces and W contains f : X → Y if the underlying map of
spaces is a weak homotopy equivalence.
(5) C is (Set ∆op ) G (the category of simplicial sets equipped with a G-action) and W is
the subcategory of weak equivalences on the underlying category Set ∆op .
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