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topology.sheaves.sheaf_condition.pairwise_intersections

Equivalent formulations of the sheaf condition #

We give an equivalent formulation of the sheaf condition.

Given any indexed type ι, we define overlap ι, a category with objects corresponding to

Any open cover U : ι → opens X provides a functor diagram U : overlap ι ⥤ (opens X)ᵒᵖ.

There is a canonical cone over this functor, cone U, whose cone point is supr U, and in fact this is a limit cone.

A presheaf F : presheaf C X is a sheaf precisely if it preserves this limit. We express this in two equivalent ways, as

We show that this sheaf condition is equivalent to the opens_le_cover sheaf condition, and thereby also equivalent to the default sheaf condition.

An alternative formulation of the sheaf condition (which we prove equivalent to the usual one below as is_sheaf_iff_is_sheaf_pairwise_intersections).

A presheaf is a sheaf if F sends the cone (pairwise.cocone U).op to a limit cone. (Recall pairwise.cocone U has cone point supr U, mapping down to the U i and the U i ⊓ U j.)

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An alternative formulation of the sheaf condition (which we prove equivalent to the usual one below as is_sheaf_iff_is_sheaf_preserves_limit_pairwise_intersections).

A presheaf is a sheaf if F preserves the limit of pairwise.diagram U. (Recall pairwise.diagram U is the diagram consisting of the pairwise intersections U i ⊓ U j mapping into the open sets U i. This diagram has limit supr U.)

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@[protected, instance]

The diagram consisting of the U i and U i ⊓ U j is cofinal in the diagram of all opens contained in some U i.

The sheaf condition in terms of a limit diagram over all { V : opens X // ∃ i, V ≤ U i } is equivalent to the reformulation in terms of a limit diagram over U i and U i ⊓ U j.

The sheaf condition in terms of an equalizer diagram is equivalent to the reformulation in terms of a limit diagram over U i and U i ⊓ U j.

The sheaf condition in terms of an equalizer diagram is equivalent to the reformulation in terms of the presheaf preserving the limit of the diagram consisting of the U i and U i ⊓ U j.

For a sheaf F, F(U ⊔ V) is the pullback of F(U) ⟶ F(U ⊓ V) and F(V) ⟶ F(U ⊓ V). This is the pullback cone.

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(Implementation). Every cone over F(U) ⟶ F(U ⊓ V) and F(V) ⟶ F(U ⊓ V) factors through F(U ⊔ V).

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For a sheaf F, F(U ⊔ V) is the pullback of F(U) ⟶ F(U ⊓ V) and F(V) ⟶ F(U ⊓ V).

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