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 #
- This is the definition Lurie uses in Spectral Algebraic Geometry.
The category of open sets contained in some element of the cover.
Equations
- Top.presheaf.sheaf_condition.opens_le_cover U = {V // ∃ (i : ι), V ≤ U i}
Instances for Top.presheaf.sheaf_condition.opens_le_cover
Equations
- Top.presheaf.sheaf_condition.opens_le_cover.category_theory.category U = category_theory.full_subcategory (λ (V : topological_space.opens ↥X), ∃ (i : ι), V ≤ U i)
An arbitrarily chosen index such that V ≤ U i.
Equations
- V.index = Exists.some _
The morphism from V to U i for some i.
Equations
supr U as a cocone over the opens sets contained in some element of the cover.
(In fact this is a colimit cocone.)
Equations
- Top.presheaf.sheaf_condition.opens_le_cover_cocone U = {X := supr U, ι := {app := λ (V : Top.presheaf.sheaf_condition.opens_le_cover U), V.hom_to_index ≫ topological_space.opens.le_supr U V.index, naturality' := _}}
An equivalent formulation of the sheaf condition
(which we prove equivalent to the usual one below as
is_sheaf_iff_is_sheaf_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.)
Equations
- F.is_sheaf_opens_le_cover = ∀ ⦃ι : Type v⦄ (U : ι → topological_space.opens ↥X), nonempty (category_theory.limits.is_limit (category_theory.functor.map_cone F (Top.presheaf.sheaf_condition.opens_le_cover_cocone U).op))
Implementation detail:
the object level of pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U
Equations
Implementation detail:
the morphism level of pairwise_to_opens_le_cover : pairwise ι ⥤ opens_le_cover U
Equations
- Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map U (category_theory.pairwise.hom.right i j) = category_theory.hom_of_le _
- Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map U (category_theory.pairwise.hom.left i j) = category_theory.hom_of_le _
- Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map U (category_theory.pairwise.hom.id_pair i j) = 𝟙 (Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_obj U (category_theory.pairwise.pair i j))
- Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map U (category_theory.pairwise.hom.id_single i) = 𝟙 (Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_obj U (category_theory.pairwise.single i))
The category of single and double intersections of the U i maps into the category
of open sets below some U i.
Equations
- Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover U = {obj := Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_obj U, map := λ (V W : category_theory.pairwise ι) (i : V ⟶ W), Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover_map U i, map_id' := _, map_comp' := _}
Instances for Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover
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 diagram in opens X indexed by pairwise intersections from U is isomorphic
(in fact, equal) to the diagram factored through opens_le_cover U.
Equations
- Top.presheaf.sheaf_condition.pairwise_diagram_iso U = {hom := {app := λ (X_1 : category_theory.pairwise ι), X_1.cases_on (λ (i : ι), 𝟙 ((category_theory.pairwise.diagram U).obj (category_theory.pairwise.single i))) (λ (i j : ι), 𝟙 ((category_theory.pairwise.diagram U).obj (category_theory.pairwise.pair i j))), naturality' := _}, inv := {app := λ (X_1 : category_theory.pairwise ι), X_1.cases_on (λ (i : ι), 𝟙 ((Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover U ⋙ category_theory.full_subcategory_inclusion (λ (V : topological_space.opens ↥X), ∃ (i : ι), V ≤ U i)).obj (category_theory.pairwise.single i))) (λ (i j : ι), 𝟙 ((Top.presheaf.sheaf_condition.pairwise_to_opens_le_cover U ⋙ category_theory.full_subcategory_inclusion (λ (V : topological_space.opens ↥X), ∃ (i : ι), V ≤ U i)).obj (category_theory.pairwise.pair i j))), naturality' := _}, hom_inv_id' := _, inv_hom_id' := _}
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.
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.
Given a family of opens U and an open Y equal to the union of opens in U, we may
take the presieve on Y associated to U and the sieve generated by it, and form the
full subcategory (subposet) of opens contained in Y (over Y) consisting of arrows
in the sieve. This full subcategory is equivalent to opens_le_cover U, the (poset)
category of opens contained in some U i.
Equations
- Top.presheaf.generate_equivalence_opens_le U hY = {functor := {obj := λ (f : {f // (category_theory.sieve.generate (Top.presheaf.presieve_of_covering_aux U Y)).arrows f.hom}), ⟨f.val.left, _⟩, map := λ (_x _x_1 : {f // (category_theory.sieve.generate (Top.presheaf.presieve_of_covering_aux U Y)).arrows f.hom}) (g : _x ⟶ _x_1), g.left, map_id' := _, map_comp' := _}, inverse := {obj := λ (V : Top.presheaf.sheaf_condition.opens_le_cover U), ⟨category_theory.over.mk _.hom, _⟩, map := λ (_x _x_1 : Top.presheaf.sheaf_condition.opens_le_cover U) (g : _x ⟶ _x_1), category_theory.over.hom_mk g _, map_id' := _, map_comp' := _}, unit_iso := category_theory.eq_to_iso _, counit_iso := category_theory.eq_to_iso _, functor_unit_iso_comp' := _}
Given a family of opens opens_le_cover_cocone U is essentially the natural cocone
associated to the sieve generated by the presieve associated to U with indexing
category changed using the above equivalence.
Equations
- F.whisker_iso_map_generate_cocone U hY = {hom := {hom := F.map (category_theory.eq_to_hom _), w' := _}, inv := {hom := F.map (category_theory.eq_to_hom _), w' := _}, hom_inv_id' := _, inv_hom_id' := _}
Given a presheaf F on the topological space X and a family of opens U of X,
the natural cone associated to F and U used in the definition of
F.is_sheaf_opens_le_cover is a limit cone iff the natural cone associated to F
and the sieve generated by the presieve associated to U is a limit cone.
Given a presheaf F on the topological space X and a presieve R whose generated sieve
is covering for the associated Grothendieck topology (equivalently, the presieve is covering
for the associated pretopology), the natural cone associated to F and the family of opens
associated to R is a limit cone iff the natural cone associated to F and the generated
sieve is a limit cone.
Since only the existence of a 1-1 correspondence will be used, the exact definition does
not matter, so tactics are used liberally.
Equations
A presheaf (opens X)ᵒᵖ ⥤ C on a topological space X is a sheaf on the site opens X iff
it satisfies the is_sheaf_opens_le_cover sheaf condition. The latter is not the
official definition of sheaves on spaces, but has the advantage that it does not
require has_products C.
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 }.