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Another side quest from @David Egolf 's thread on reading Baez topos series.
Let be an arbitrary topological space.
First, we defined a functor from the category of presheaves over to the category of bundles over . maps a presheaf over to an étale space over .
Second, we defined a functor . maps a bundle to its sheaf of sections over .
I was wondering if there were an adjunction .
I managed to get a proof that, indeed, is left adjoint to . But my proof is relatively long, and I am not sure I checked everything right. An external confirmation/refutation is welcome.
ps: According to John in the other thread, the composite is called "sheafification", and appears as the left adjoint to the forgetful functor from sheaves to presheaves. This led me to wonder if sheafification were a monad.
Peva Blanchard said:
I managed to get a proof that, indeed, is left adjoint to . But my proof is relatively long, and I am not sure I checked everything right. An external confirmation/refutation is welcome.
Was your proof too long to fit in the margins of this stream? :P
(Seriously, where do I find it? I'd like to read it.)
Peva Blanchard said:
I was wondering if there were an adjunction .
Try the course notes: the end of Part 2 and also the section "The adjunction between presheaves and bundles" in Part 3.
This led me to wonder if sheafification were a monad.
Since it's a left adjoint followed by a right adjoint, it's a monad. But even better, I believe it's an [[idempotent monad]]. That is, if you sheafify a presheaf that's already a sheaf, you don't get anything new, just something isomorphic to what you had.
John Baez said:
Peva Blanchard said:
I was wondering if there were an adjunction .
Try the course notes: the end of Part 2 and also the section "The adjunction between presheaves and bundles" in Part 3.
Oops ! I should have known the answer was already there. I read the series last year: perhaps I was just remembering the adjunction without realizing that I read about it there.
Anyway, thank you for the link.
Eric M Downes said:
Peva Blanchard said:
I managed to get a proof that, indeed, is left adjoint to . But my proof is relatively long, and I am not sure I checked everything right. An external confirmation/refutation is welcome.
Was your proof too long to fit in the margins of this stream? :P
(Seriously, where do I find it? I'd like to read it.)
Yes sure :)
I chose to prove the adjunction by building a natural isomorphism
where is a presheaf over , and a bundle over .
Some notations. Given a section over an open subset , is the germ of at , and is the continuous function . We also denote by the set of germs of over , which is a member of a base of open sets of .
Consider a bundle morphism , i.e., a continuous function that preserves the base point. We want to define a natural transformation from the presheaf to the sheaf of sections of .
Let be an open subset of , and a section of over .
We want to define a section of over , i.e., a continuous function such that . For every , let's define
Because is a morphism of bundles over , we have .
It remains to show that is continuous. By construction of , we know that the map
is continuous. By the continuity of and the fact that the composition of two continuous functions is continuous, we conclude that is also continuous.
To sum up, we defined a function
Consider an inclusion . We want to show that the following diagram commutes.
image.png
This amounts to show that
Therefore, we have a natural transformation .
Let be a natural transformation. We want to define a morphism of bundles, i.e. a continuous function that preserves the base point.
Fix a germ , for some section over an open neighborhood of . We define
We need to show that is well-defined. Assume for another section over another open neighborhood of . Without loss of generality, we may assume that . In that case . Then
Therefore, is well-defined. Note also that, by definition, preserves the base point.
Let be an open neighborhood of in . Without loss of generality, we may, again, assume that . Then,
is the restriction of on .
Consider , and the open neighborhood of . For every ,
In other words,
I.e., is continuous.
We have shown that
and
are both well-defined functions.
Let be a morphism of bundles over , and .
By definition, for every for some open neighborhood of ,
I.e., , and thus .
Let be a natural transformation of presheaves, and .
By definition, for every open subset of ,
I.e., , and thus .
We show that transforms naturally with respect to .
Consider a morphism of bundles .
The naturality condition is expressed in the following diagram
I.e., we want to show that, for every open subset ,
I.e., for every ,
which holds by the associativity.
We show that transforms naturally with respect to .
Consider a natural transformation of presheaves over . The naturality condition is expressed in the following diagram
image.png
I.e., we want to show that, for every open subset ,
I.e., for every section ,
We have shown that every component is the inverse of the component (as functions in ). Because is natural in and , I think we can conclude that is also natural in and .
This concludes the proof that is left adjoint to .
Wow, great! When @David Egolf reaches this material in the course, he can either try to figure out himself why is left adjoint to and check his work here, or just read this!