mathlib documentation

category_theory.sites.closed

Closed sieves #

A natural closure operator on sieves is a closure operator on sieve X for each X which commutes with pullback. 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 J, and further this presheaf can be used to determine the Grothendieck topology from the sheaf predicate. 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 #

Tags #

closed sieve, closure, Grothendieck topology

References #

The J-closure of a sieve is the collection of arrows which it covers.

Equations
@[simp]

Any sieve is smaller than its closure.

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.

Equations

If S is J₁-closed, then S covers exactly the arrows it contains.

Being J-closed is stable under pullback.

The closure of a sieve S is the largest closed sieve which contains S (justifying the name "closure").

The closure of a sieve is closed.

The sieve S is closed iff its closure is equal to itself.

The sieve 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.

Equations

The presheaf sending each object to the set of J-closed sieves on it. This presheaf is a J-sheaf (and will turn out to be a subobject classifier for the category of J-sheaves).

Equations

The presheaf of J-closed sieves is a J-sheaf. The proof of this is adapted from [MM92], Chatper III, Section 7, Lemma 1.

If presheaf of J₁-closed sieves is a J₂-sheaf then J₁ ≤ J₂. Note the converse is true by classifier_is_sheaf and is_sheaf_of_le.

If being a sheaf for J₁ is equivalent to being a sheaf for J₂, then 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.

Equations

The topology given by the closure operator J.close on a Grothendieck topology is the same as J.