alternative lecture notes - Rational points and algebraic cycles

Proof. It suffices to check that const preserves fibrations and trivial fibrations. This follows
simply from the definition of projective model structure. If f is a fibration, then const(f)
is level-wise just f. Since it is level-wise a fibration, it is a fibration in the projective model
structure. Similarly for trivial fibrations.
□
Dually, the functors const and lim define a Quillen adjunction between M and M D endowed
with the injective model structure:
const: M
M D : lim
Theorem 11.2. Let C, D be Grothendieck sites and f : D → C a morphism of sites. Let
Sh(D) ∆op and Sh(C) ∆op be the categories of simplicial sheaves of abelian groups on D and
C, both endowed with the Joyal model structure. Then
is a Quillen adjunction.
f ∗ : Sh(D) ∆op
Sh(C) ∆op : f ∗
Example 11.3. Let D be the trivial site (i.e. the Zariski site of the empty set). Then Sh(D)
is Set. Theorem 11.2 says that, for any site C, we get a Quillen adjunction
Γ ∗ : Set ∆op
Sh(C) ∆op : Γ ∗
where Γ ∗ = const is taking the constant sheaf and Γ ∗ is taking global sections.
Example 11.4. Let D be as in Example 11.3. Let C be a category endowed with the trivial
Grothendieck topology. This means that the covering families are the isomorphisms. The
sheaf condition now becomes vacuous. Then all presheaves are sheaves, so that Sh(C) =
Set C , and Sh(C) ∆op = (Set ∆op ) C . Now the Joyal model structure on Sh(C) ∆op coincides
with the injective model structure on (Set ∆op ) C . The Quillen adjunction takes the form
const: Set ∆op
(Set ∆op ) C : lim
where the functor on the right is taking the limit over C.
We briefly explain why the Joyal model structure and injective model structure on Sh(C) ∆op
coincide when C has the trivial topology. For this, we may check that the cofibrations and
weak equivalences coincide. In both model structures, the cofibrations are the levelwise and
objectwise injective maps. One can further show that the topology being trivial implies that
the level-wise weak equivalences coincide with the stalk-wise weak equivalences.
Example 11.5. Let M be a simplicial model category and let C ∈ M be a cofibrant object.
Then there is a Quillen adjunction between SSet and M:
• × C : SSet
M : Map(C, •)
This is simply a restatement of the fact that M is copowered over SSet.
Dually:
Example 11.6. Let M be a simplicial model category and let F ∈ M be a fibrant object.
Then there is a Quillen adjunction between SSet and the opposite category of M:
F • : SSet
M op : Map(•, F )
This is simply a restatement of the fact that M is powered over SSet.
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