# mathlibdocumentation

topology.sheaves.sheaf_condition.opens_le_cover

# Another version of the sheaf condition. #

Given a family of open sets U : ι → opens X we can form the subcategory { V : opens X // ∃ i, V ≤ U i }, which has supr U as a cocone.

The sheaf condition on a presheaf F is equivalent to F sending the opposite of this cocone to a limit cone in C, for every U.

This condition is particularly nice when checking the sheaf condition because we don't need to do any case bashing (depending on whether we're looking at single or double intersections, or equivalently whether we're looking at the first or second object in an equalizer diagram).

## References #

def Top.presheaf.sheaf_condition.opens_le_cover {X : Top} {ι : Type v} (U : ι → ) :
Type v

The category of open sets contained in some element of the cover.

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• = {V // ∃ (i : ι), V U i}
@[instance]
def Top.presheaf.sheaf_condition.opens_le_cover.inhabited {X : Top} {ι : Type v} (U : ι → ) [inhabited ι] :
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@[instance]
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def Top.presheaf.sheaf_condition.opens_le_cover.index {X : Top} {ι : Type v} {U : ι → }  :
ι

An arbitrarily chosen index such that V ≤ U i.

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def Top.presheaf.sheaf_condition.opens_le_cover.hom_to_index {X : Top} {ι : Type v} {U : ι → }  :
V.val U V.index

The morphism from V to U i for some i.

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supr U as a cocone over the opens sets contained in some element of the cover.

(In fact this is a colimit cocone.)

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def Top.presheaf.is_sheaf_opens_le_cover {C : Type u} {X : Top} (F : X) :
Prop

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

A presheaf is a sheaf if F sends the cone (opens_le_cover_cocone U).op to a limit cone. (Recall opens_le_cover_cocone U, has cone point supr U, mapping down to any V which is contained in some U i.)

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@[simp]
def Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_obj {X : Top} {ι : Type v} (U : ι → ) :

Implementation detail: the object level of pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U

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Implementation detail: the morphism level of pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U

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@[simp]
theorem Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map_2 {X : Top} {ι : Type v} (U : ι → ) (V W : category_theory.pairwise ι) (i : V W) :
def Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover {X : Top} {ι : Type v} (U : ι → ) :

The category of single and double intersections of the U i maps into the category of open sets below some U i.

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@[simp]
theorem Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_obj_2 {X : Top} {ι : Type v} (U : ι → ) (ᾰ : category_theory.pairwise ι) :
@[instance]
@[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.

def Top.presheaf.sheaf_condition.pairwise_diagram_iso {X : Top} {ι : Type v} (U : ι → ) :

The diagram in opens X indexed by pairwise intersections from U is isomorphic (in fact, equal) to the diagram factored through opens_le_cover U.

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The cocone pairwise.cocone U with cocone point supr U over pairwise.diagram U is isomorphic to the cocone opens_le_cover_cocone U (with the same cocone point) after appropriate whiskering and postcomposition.

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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.

theorem Top.presheaf.is_sheaf_iff_is_sheaf_opens_le_cover {C : Type u} {X : Top} (F : X) :

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