Room for Arrow and Hanno

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Arrow

16:14

@Hanno hello!
I was wondering whether you had a minute or two..

I'm trying to understand, concretely, in terms of bundles of sets, what the dependent product is.

In particular, how should I imagine local Cartesian closedness of the category of sets geometrically

I understand it should relate to sections as "generalized functions" with varying codomains

Hanno

16:41

hey

Arrow

Hello :)

I think I figured out what I wanted to ask. Is it okay if I ask something different?

Hanno

If you have a function f with domain some set X, you may think of f as a family of sets varying over x. the dependent product is then just prod_{x\in X} f(x)

now you just have to make the translation from families of sets to maps of sets

as usual: a map f: Z -> X is viewed as a family of sets parametrized over X by sending x\in X to its preimage

so an element of dep(f) is a function assigning to each x of X an element of Z lying over x

sure @ ask sth diff, what do you want to know?

Arrow

My motivation is just trying to tidy up some notes of mine on Hurewicz connection. My general question is here math.stackexchange.com/questions/2121027/…

But the particular question I have is whether base change along the cylinder evaluation $\mathrm{ev}_0:B^I\to B$ admits a right adjoint.

This is probably boring technical stuff though...

Hanno

uh.. yea sounds technical at first. did you try just mimicking the set construction of the dependent product in the category of k spaces?

Arrow

I couldn't find it in May's book

It's terse and I don't really understand anything. I Don't know of any elementary accounts of dependent products of spaces anywhere.

By the way, if the proposition is true, then base change along $\mathrm{ev}_0^B$ has a right adjoint as soon as $B$ is compactly generated, right? Since anything can be exponentiated by the unit interval (?)

Hanno

16:51

I think so

Arrow

Cool. Then I can described maps out of the mapping cocylinder of bundles with a $k$-space base using the dependent product :)

Hanno

I'm sorry I don't think I'm the best person to ask for this stuff. Maybe Mike or Tim will answer to your comment-question on MO?

Arrow

I hope so! Sorry for bugging you uselessly, I didn't expect to figure out the dependent stuff myself.

Hanno

no worries

Arrow