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:
A sieve S
on X
is covering for finestTopologySingle P
iff
for any f : Y ⟶ X
, P
satisfies the sheaf axiom for S.pullback f
.
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.
Using 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.
References #
- https://ncatlab.org/nlab/show/canonical+topology
- https://ncatlab.org/nlab/show/subcanonical+coverage
- https://stacks.math.columbia.edu/tag/00Z9
- https://math.stackexchange.com/a/358709/
To show P
is a sheaf for the binding of U
with B
, it suffices to show that P
is a sheaf for
U
, that P
is a sheaf for each sieve in B
, and that it is separated for any pullback of any
sieve in B
.
This is mostly an auxiliary lemma to show isSheafFor_trans
.
Adapted from [Elephant], Lemma C2.1.7(i) with suggestions as mentioned in
https://math.stackexchange.com/a/358709/
Given two sieves R
and S
, to show that P
is a sheaf for S
, we can show:
P
is a sheaf forR
P
is a sheaf for the pullback ofS
along any arrow inR
P
is separated for the pullback ofR
along any arrow inS
.
This is mostly an auxiliary lemma to construct finestTopology
.
Adapted from [Elephant], Lemma C2.1.7(ii) with suggestions as mentioned in
https://math.stackexchange.com/a/358709
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).
Instances For
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.
Instances For
Check that if P ∈ Ps
, then P
is indeed a sheaf for the finest topology on Ps
.
Check that if each P ∈ Ps
is a sheaf for J
, then J
is a subtopology of finestTopology Ps
.
The canonicalTopology
on a category is the finest (largest) topology for which every
representable presheaf is a sheaf.
See https://stacks.math.columbia.edu/tag/00ZA
Instances For
yoneda.obj X
is a sheaf for the canonical topology.
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.
Instances For
If every functor yoneda.obj X
is a J
-sheaf, then J
is subcanonical.
If J
is subcanonical, then any representable is a J
-sheaf.