The sheaf condition in terms of unique gluings #
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We provide an alternative formulation of the sheaf condition in terms of unique gluings.
We work with sheaves valued in a concrete category C admitting all limits, whose forgetful
functor C ⥤ Type preserves limits and reflects isomorphisms. The usual categories of algebraic
structures, such as Mon, AddCommGroup, Ring, CommRing etc. are all examples of this kind of
category.
A presheaf F : presheaf C X satisfies the sheaf condition if and only if, for every
compatible family of sections sf : Π i : ι, F.obj (op (U i)), there exists a unique gluing
s : F.obj (op (supr U)).
Here, the family sf is called compatible, if for all i j : ι, the restrictions of sf i
and sf j to U i ⊓ U j agree. A section s : F.obj (op (supr U)) is a gluing for the
family sf, if s restricts to sf i on U i for all i : ι
We show that the sheaf condition in terms of unique gluings is equivalent to the definition
in terms of equalizers. Our approach is as follows: First, we show them to be equivalent for
Type-valued presheaves. Then we use that composing a presheaf with a limit-preserving and
isomorphism-reflecting functor leaves the sheaf condition invariant, as shown in
topology/sheaves/forget.lean.
A family of sections sf is compatible, if the restrictions of sf i and sf j to U i ⊓ U j
agree, for all i and j
Equations
- F.is_compatible U sf = ∀ (i j : ι), ⇑(F.map ((U i).inf_le_left (U j)).op) (sf i) = ⇑(F.map ((U i).inf_le_right (U j)).op) (sf j)
A section s is a gluing for a family of sections sf if it restricts to sf i on U i,
for all i
The sheaf condition in terms of unique gluings. A presheaf F : presheaf C X satisfies this sheaf
condition if and only if, for every compatible family of sections sf : Π i : ι, F.obj (op (U i)),
there exists a unique gluing s : F.obj (op (supr U)).
We prove this to be equivalent to the usual one below in
is_sheaf_iff_is_sheaf_unique_gluing
Equations
- F.is_sheaf_unique_gluing = ∀ ⦃ι : Type v⦄ (U : ι → topological_space.opens ↥X) (sf : Π (i : ι), ↥(F.obj (opposite.op (U i)))), F.is_compatible U sf → (∃! (s : ↥(F.obj (opposite.op (supr U)))), F.is_gluing U sf s)
For presheaves of types, terms of pi_opens F U are just families of sections.
Equations
- F.pi_opens_iso_sections_family U = (category_theory.limits.limit.is_limit (category_theory.discrete.functor (λ (i : ι), F.obj (opposite.op (U i))))).cone_point_unique_up_to_iso (category_theory.limits.types.product_limit_cone (λ (i : ι), F.obj (opposite.op (U i)))).is_limit
Under the isomorphism pi_opens_iso_sections_family, compatibility of sections is the same
as being equalized by the arrows left_res and right_res of the equalizer diagram.
Under the isomorphism pi_opens_iso_sections_family, being a gluing of a family of
sections sf is the same as lying in the preimage of res (the leftmost arrow of the
equalizer diagram).
The "equalizer" sheaf condition can be obtained from the sheaf condition in terms of unique gluings.
The sheaf condition in terms of unique gluings can be obtained from the usual "equalizer" sheaf condition.
For type-valued presheaves, the sheaf condition in terms of unique gluings is equivalent to the usual sheaf condition in terms of equalizer diagrams.
For presheaves valued in a concrete category, whose forgetful functor reflects isomorphisms and preserves limits, the sheaf condition in terms of unique gluings is equivalent to the usual one in terms of equalizer diagrams.
A more convenient way of obtaining a unique gluing of sections for a sheaf.
In this version of the lemma, the inclusion homs iUV can be specified directly by the user,
which can be more convenient in practice.
In this version of the lemma, the inclusion homs iUV can be specified directly by the user,
which can be more convenient in practice.