alternative lecture notes - Rational points and algebraic cycles

Define the nerve N ∆ ((U α )) of a covering U := ∐ α∈A U α → X as the simplicial set with
⎛
⎞
N ∆ ((U α )) n := π 0
⎜
⎝ U × · · · × U⎟
X X
⎠ .
} {{ }
n+1
Given f, g : X → Y , we say that f ∼ g if there exists H : I × X → Y such that H| 0 = f
and H| 1 = g.
Define the simplicial set ∆ n : ∆ op → Set by ∆ n ([m]) := Hom ∆ ([m], [n]). For example, ∆ 0
is a point.
Definition 5.12. Let A and B be simplicial sets, and let f : A → B. We say that f ∼ st
g
if there exists H : A × ∆ 1 → B such that H| 0 = f and H| 0 = g. This is reflexive, but not
symmetric or transitive. Define the relation ∼ as the equivalence relation generated by ∼.
st
Definition 5.13. Let A and B be simplicial sets. Define the mapping space Map(A, B) ∈
Set ∆op
as the simplicial set given by Map(A, B) n := Hom Set ∆ op (∆ n × A, B).
For X, Y, Z ∈ CGHS, we have the exponential law
Map(X × Y, Z) ≈ Map(Y, Map(X, Z)).
Let A, B be simplicial sets. Let f, g : A → B. The following are equivalent:
(1) f ∼ g
(2) f and g (or rather, the realization of the corresponding two points in Map(A, B) 0 )
lie in the same connected component of | Map(A, B)|.
If A and B are simplicial sets, there is a map
φ: | Map(A, B)| → Map Top (|A|, |B|).
The map from a circle to a circle that wraps around twice does not come from a simplicial
map (unless one subdivides). Dan Kan solved this problem as follows.
Definition 5.14. Let Λ n k be ∆n minus the interior and k th face. A Kan simplicial set is a
simplicial set A such that for every diagram of the form
the diagonal map exists.
Λ n k
∆ n
A
∃
Theorem 5.15. If B is Kan, then φ above is a homotopy equivalence.
6. July 18 (Harpaz)
If C and D are categories with finite limits, and F : C → D is a functor that preserves
them, then Pro(F ): Pro(C) → Pro(D) preserves all limits.
The inclusion functor FinGr → Groups preserves only finite limits, but we get an adjoint
P L: Pro(Groups) → Pro(FinGr).
12