alternative lecture notes - Rational points and algebraic cycles

Example 11.11. (Colimits do not preserve weak equivalences in general.) Let M = SSet
and D be the category defined by the following graph
d 1
d 2
We consider M D with the projective model structure. Let A = (A 1 , A 2 , A 3 ) ∈ M D be the
object defined by A 1 = {pt ′ , pt ′′ }, A 2 = A 3 = pt. The colimit of A is a single point. Let
object B = (B 1 , B 2 , B 3 ) ∈ M D be defined by B 1 = {pt ′ , pt ′′ }, B 2 = B 3 = line segment,
where the non-identity maps send pt ′ , pt ′′ to the end points of both line segments. The
colimit of B is a circle. There is an obvious element of Hom M D(B, A) that collapses the
line segments to points. This map is a weak equivalence, since it is a weak equivalence
levelwise. The induced map colim B → colim A collapses a circle onto a point, so is not a
weak equivalence. Therefore, colim does not preserve weak equivalences in general.
Exercise: Let A be an object in M D . If A 1 is a cofibrant object and A 1 → A 2 and
A 1 → A 3 are cofibrations in M, then A is cofibrant in M D with the projective model
structure. (Hint: use that the cofibrations are exactly the morphisms satisfying the LLP
with respect to trivial fibrations.)
From the exercise it follows that the colimit of the B of Example 11.11 is in fact the
homotopy colimit.
Example 11.12. Here is how to take the cofibrant replacement A cof of A as in Example
11.11. Set A cof
1 = A 1 , and define A cof
2 and A cof
3 respectively as the mapping cylinders of
A 1 → A 2 and A 1 → A 3 . There are canonical inclusions A cof
1 → A cof
2 and A cof
1 → A cof
3 .
Taking the colimit can be described geometrically as gluing the two mapping cylinders along
the images of A 1 .
Dually, if B is a diagram in M D , the homotopy limit turns out to be
holim(B) = B 2 × h B 1
B 3
= {(b 2 , b 3 , p) : b 2 ∈ B 2 , b 3 ∈ B 3 , p : [0, 1] → B 1 : p(0) = b 2 , p(1) = b 3 } .
Let G be a group, considered as a category with one object. Then we have the Quillen
adjunction
colim: SSet G
SSet: const
The colimit maps a simplicial G-set to its orbit space. Recall that an object of SSet G is
cofibrant if and only if the action is free. To compute the homotopy colimit of X, one
replaces X by X ×EG, which is a cofibrant replacement since EG is contractible. Hence the
homotopy colimit of X is weakly equivalent to (X × EG)/G. This is the Borel construction
which we encountered in Section 2. When X is a point with trivial G-action, we get that
the homotopy colimit of X is BG, the classifying space of G.
Dually, we have:
const: SSet SSet G : lim
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d 3