The canonical topology on a category #
We define the finest (largest) Grothendieck topology for which a given presheaf
P is a sheaf.
This is well defined since if
P is a sheaf for a topology
J, then it is a sheaf for any
coarser (smaller) topology. Nonetheless we define the topology explicitly by specifying its sieves:
X is covering for
finest_topology_single P iff
f : Y ⟶ X,
P satisfies the sheaf axiom for
Showing that this is a genuine Grothendieck topology (namely that it satisfies the transitivity
axiom) forms the bulk of this file.
This generalises to a set of presheaves, giving the topology
finest_topology Ps which is the
finest topology for which every presheaf in
Ps is a sheaf.
Ps as the set of representable presheaves defines the
canonical_topology: the finest
topology for which every representable is a sheaf.
A Grothendieck topology is called
subcanonical if it is smaller than the canonical topology,
equivalently it is subcanonical iff every representable presheaf is a sheaf.
P is a sheaf for the binding of
B, it suffices to show that
P is a sheaf for
P is a sheaf for each sieve in
B, and that it is separated for any pullback of any
Given two sieves
S, to show that
P is a sheaf for
S, we can show:
Pis a sheaf for
Pis a sheaf for the pullback of
Salong any arrow in
Pis separated for the pullback of
Ralong any arrow in
Construct the finest (largest) Grothendieck topology for which the given presheaf is a sheaf.
This is a special case of https://stacks.math.columbia.edu/tag/00Z9, but following a different proof (see the comments there).
Construct the finest (largest) Grothendieck topology for which all the given presheaves are sheaves.
This is equal to the construction of https://stacks.math.columbia.edu/tag/00Z9.
Check that if
P ∈ Ps, then
P is indeed a sheaf for the finest topology on
Check that if each
P ∈ Ps is a sheaf for
J is a subtopology of
canonical_topology on a category is the finest (largest) topology for which every
representable presheaf is a sheaf.
A subcanonical topology is a topology which is smaller than the canonical topology. Equivalently, a topology is subcanonical iff every representable is a sheaf.
If every functor
yoneda.obj X is a
J is subcanonical.
J is subcanonical, then any representable is a