alternative lecture notes - Rational points and algebraic cycles

Theorem 4.11. If C has finite limits, then Pro(C) has all limits, i.e., it is complete.
Proof. Any limit can be expressed as an equalizer (limit of diagram A ⇒ B) of a pair of
morphisms between products: specifically, lim F (i) is the equalizer of
∏
F (i) ⇒
∏
F (i ′ )
i∈I
ρ=(i→i ′ )∈I
in which the first morphism is given by of ∏ i∈I F (i) pr i
−→ ′
F (i ′ ), and the second morphism is
given by of ∏ i∈I F (i) pr i
−→ F (i) −→ F (ρ)
F (i ′ ).
Thus it is enough to show that Pro(C) has arbitrary products (over any index set) and
equalizers. Since Pro(C) has finite limits, it has finite products and equalizers. Arbitrary
products are filtered limits of finite products:
∏
∏
A i =
A i .
i∈P
lim
finite S ⊆ P
Further claim: If C and D have finite limits and F : C → D preserves finite limits, then
Pro(F ): Pro(C) → Pro(D) preserves all limits.
Recall from last lecture that a functor preserving all limits (modulo smallness conditions)
has a left adjoint. So in the situation above, we obtain a left adjoint P L: Pro(D) → Pro(C)
of Pro(F ).
i∈S
5. July 16 (Schlank)
Today: Combinatorial and categorical approach to homotopy theory
Example 5.1. A circle, annulus, and solid torus are homotopy equivalent.
Homotopy theory is the study of the ways that contractible spaces can be glued together
to give global objects.
Definition 5.2. Let X be a space and let (U α ) φ∈A be a cover of X. Say that (U α ) is excellent
if for every finite nonempty subset I ⊆ A, the intersection U I := ⋂ i∈I U i is either contractible
or empty.
Example 5.3. The circle has an excellent cover given by three overlapping half-circles.
Proposition 5.4. Let X be a paracompact space. Let (U α ) be an excellent cover of X. Then
X is homotopically equivalent to the nerve of (U α ), where the nerve is the space constructed
by
(1) taking a point for every U α ,
(2) connecting U α and U β by a line segment if U α ∩ U β ≠ ∅,
(3) filling in a triangle if U α ∩ U β ∩ U γ ≠ ∅,
(4) etc.,
with the colimit topology of the Euclidean topologies on the simplices.
The circle, annulus, and solid torus have combinatorially equivalent excellent covers, so
they are homotopically equivalent.
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