Closed sieves #
A natural closure operator on sieves is a closure operator on
sieve X for each
X which commutes
We show that a Grothendieck topology
J induces a natural closure operator, and define what the
closed sieves are. The collection of
J-closed sieves forms a presheaf which is a sheaf for
and further this presheaf can be used to determine the Grothendieck topology from the sheaf
Finally we show that a natural closure operator on sieves induces a Grothendieck topology, and hence
that natural closure operators are in bijection with Grothendieck topologies.
Main definitions #
category_theory.grothendieck_topology.close: Sends a sieve
Xto the set of arrows which it covers. This has all the usual properties of a closure operator, as well as commuting with pullback.
category_theory.grothendieck_topology.closure_operator: The bundled
category_theory.grothendieck_topology.closed: A sieve
Xis closed for the topology
Jif it contains every arrow it covers.
category_theory.functor.closed_sieves: The presheaf sending
Xto the collection of
J-closed sieves on
X. This is additionally shown to be a sheaf for
J, and if this is a sheaf for a different topology
J' ≤ J.
category_theory.grothendieck_topology.topology_of_closure_operator: A closure operator on the set of sieves on every object which commutes with pullback additionally induces a Grothendieck topology, giving a bijection with
closed sieve, closure, Grothendieck topology
J-closure of a sieve is the collection of arrows which it covers.
A sieve is closed for the Grothendieck topology if it contains every arrow it covers. In the case of the usual topology on a topological space, this means that the open cover contains every open set which it covers.
Note this has no relation to a closed subset of a topological space.
S covers exactly the arrows it contains.
J-closed is stable under pullback.
The closure of a sieve
S is the largest closed sieve which contains
S (justifying the name
J is stable under pullback.
S is in the topology iff its closure is the maximal sieve. This shows that the closure
operator determines the topology.
A Grothendieck topology induces a natural family of closure operators on sieves.
The presheaf sending each object to the set of
J-closed sieves on it. This presheaf is a
(and will turn out to be a subobject classifier for the category of
The presheaf of
J-closed sieves is a
The proof of this is adapted from [MM92], Chatper III, Section 7, Lemma 1.
If presheaf of
J₁-closed sieves is a
J₁ ≤ J₂. Note the converse is true by
If being a sheaf for
J₁ is equivalent to being a sheaf for
J₁ = J₂.
A closure (increasing, inflationary and idempotent) operation on sieves that commutes with pullback induces a Grothendieck topology. In fact, such operations are in bijection with Grothendieck topologies.
The topology given by the closure operator
J.close on a Grothendieck topology is the same as