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Stream: deprecated: topos theory

Topic: subobject classifiers in sheaves

sarahzrf (Apr 08 2020 at 16:25):

What's the étalé space of Ω in Sh(X)? I think I thought it was the trivial bundle w/ fiber Sierpinski space, but now i'm starting to confuse myself

sarahzrf (Apr 08 2020 at 16:26):

consider Sh(R) and the fiber over 0 in the étalé space for Ω

sarahzrf (Apr 08 2020 at 16:27):

OH WAIT

sarahzrf (Apr 08 2020 at 16:28):

is it that Ω is the sheaf of sections of that bundle, but that bundle is not actually étalé

sarahzrf (Apr 08 2020 at 16:28):

/me tries to turn on point-set topology mode

sarahzrf (Apr 08 2020 at 16:31):

augh right ok, any basic open containing a point w/ fiber coordinate 0 is gonna be one on which the projection is not injective

sarahzrf (Apr 08 2020 at 16:31):

:face_palm:

sarahzrf (Apr 08 2020 at 16:32):

of course i noticed that as soon as i actually made a post

Morgan Rogers (he/him) (Apr 08 2020 at 16:33):

I'm struggling to follow your train of thought, but my experience is that $\Omega$ is not a particularly nice thing to work with as a bundle.

sarahzrf (Apr 08 2020 at 16:34):

well, so, if S is the sierpinski space, then i think maybe the sheaf of sections of π₂ : S × X → X is indeed gonna be [isomorphic to] Sh(X)'s Ω

Morgan Rogers (he/him) (Apr 08 2020 at 16:34):

As a sheaf, for each open set $U$ , $\Omega(U)$ is the set of opens contained in $U$ , which for $\mathbb{R}$ is always a lot

sarahzrf (Apr 08 2020 at 16:35):

sure, but as an étalé space, the fiber over a given point is just the germs

sarahzrf (Apr 08 2020 at 16:36):

at first i was expecting that to collapse down to sierpinski space, at least if your space has mild separation properties

sarahzrf (Apr 08 2020 at 16:36):

like i was thinking maybe a germ of a set is "whether the point is in it"

sarahzrf (Apr 08 2020 at 16:36):

and it took me a while to notice that if you're on the boundary of a set, you have a different germ from other the empty set or the whole space...

Morgan Rogers (he/him) (Apr 08 2020 at 16:36):

sarahzrf said:

sure, but as an étalé space, the fiber over a given point is just the germs

which would be... the filters of opens which contain $x$ , up to the equivalence relation of eventual equality? :grimacing:

sarahzrf (Apr 08 2020 at 16:37):

well ok

Morgan Rogers (he/him) (Apr 08 2020 at 16:37):

sarahzrf said:

like i was thinking maybe a germ of a set is "whether the point is in it"

These are the germs of the sheaves corresponding to the open subsets :+1:

sarahzrf (Apr 08 2020 at 16:38):

you mean like the representable ones?

Morgan Rogers (he/him) (Apr 08 2020 at 16:38):

Yes, those

sarahzrf (Apr 08 2020 at 16:38):

Morgan Rogers (he/him) (Apr 08 2020 at 16:39):

It's at times like this that I'm grateful for the duality between etale bundles and sheaves :upside_down:

sarahzrf (Apr 08 2020 at 16:39):

^_^

sarahzrf (Apr 08 2020 at 16:39):

3 cheers for idempotent adjunctions

edit to add: fact worth securing in the back of your mind if it's not already: all adjunctions of posets (and hence all monads on posets) are idempotent :-)