mathlib documentation

topology.sheaves.sheaf_of_functions

Sheaf conditions for presheaves of (continuous) functions.

We show that

For

Future work

Obviously there's more to do:

def Top.presheaf.to_Types (X : Top) (T : X → Type u) :

We show that the presheaf of functions to a type T (no continuity assumptions, just plain functions) form a sheaf.

In fact, the proof is identical when we do this for dependent functions to a type family T, so we do the more general case.

Equations
def Top.presheaf.to_Type (X : Top) (T : Type u) :

The presheaf of not-necessarily-continuous functions to a target type T satsifies the sheaf condition.

Equations
def Top.sheaf_to_Types (X : Top) :
(X → Type u)Top.sheaf (Type u) X

The sheaf of not-necessarily-continuous functions on X with values in type family T : X → Type u.

Equations
def Top.sheaf_to_Type (X : Top) :
Type uTop.sheaf (Type u) X

The sheaf of not-necessarily-continuous functions on X with values in a type T.

Equations