alternative lecture notes - Rational points and algebraic cycles

A relative functor F : (D, W 1 ) → (C, W 2 ) is a functor D → C such that F (W 1 ) ⊆
W 2 . Denote by RelCat the category of relative categories. There is a forgetful functor
F : RelCat → Cat (forget W ). There is also a functor U : Cat → RelCat sending C to
(C, Iso(C)). The functor U has a left adjoint Ho: RelCat → Cat:
Hom RelCat ((C, W ), U(D)) = Hom Cat (Ho(C), D)
(Here Ho(C) means Ho(C, W ).) Another notation: C[W −1 ] = Ho(C), because it is the
universal category in which the morphisms of W become isomorphisms.
We get equivalences of categories Ho(Set ∆op ) ↔ Ho(Top), but something is lost: these
categories do not have limits and colimits. Also, passing to the homotopy category forgets
too much: the morphisms in this category are just π 0 (Map(X, Y )), but we may be interested
in π 1 (Map(X, Y )), etc.
Let TopCat be the category of categories whose Hom sets are topological spaces and such
that composition is continuous.
Dwyer–Kan localization of a relative category (also called hammock localization): We define
L H : RelCat → TopCat, sending C to a category L H C with the same objects and whose
morphisms between a, b ∈ C are a simplicial set Map L H
C
(a, b). Take the set of zigzags
a ← c 1 → c 2 ← · · · c n → b
where all the left arrows are in W up to equivalence, where equivalence includes
• reversing isomorphisms (replacing an isomorphism by its inverse in the other direction)
• replacing pairs of consecutive maps by their composition (and vice versa)
• adding or removing identity morphisms
An edge is an equivalence class of diagrams
a
c 0
W
c 1
W
· · · c n
W b
c ′ 0 c ′ 1
· · · c ′ n
in which the directions of the arrows in the top row are arbitrary but match the directions
in the bottom row. A 2-simplex is an equivalence class of diagrams obtained by “composing”
two edges (to form a wider hammock), and so on.
We have π 0 (L H C ) ≃ Ho(C). (Here π 0(L H C ) is the category with the same objects, but in
which each Hom space has been replaced by its set of connected components.)
The space Map L H
C
(a, b) is called the derived mapping space. There is a map of sets
Hom C (a, b) → π 0 (Map L H
C
(a, b)).
Definition 7.2 (Quillen 1969). A model category is (C, W, FIB, COF) where
(1) C is a complete co-complete category,
(2) W , FIB, COF are subcategories with the same objects as C and containing the
subcategory of isomorphisms
(3) W satisfies the 2-out-of-3 property
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