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.

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    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.

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

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

      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.

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        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).

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          The presheaf of J-closed sieves is a J-sheaf. The proof of this is adapted from [MM92], Chapter 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_isSheaf and isSheaf_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.

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