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
finestTopologySingle 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
finestTopology Ps which is the
finest topology for which every presheaf in
Ps is a sheaf.
Ps as the set of representable presheaves defines the
canonicalTopology: 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
This is mostly an auxiliary lemma to show
Adapted from [Elephant], Lemma C2.1.7(i) with suggestions as mentioned in
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
This is mostly an auxiliary lemma to construct
Adapted from [Elephant], Lemma C2.1.7(ii) with suggestions as mentioned 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
canonicalTopology on a category is the finest (largest) topology for which every
representable presheaf is a sheaf.
A representable functor is a sheaf for the canonical topology.
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