Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: something like a slice category?

Tim Hosgood (Jan 26 2021 at 03:35):

a simple question which my poor brain cannot unravel at the moment: what is the sleek categorical definition of the following category?

Given a topological space $B$, let $\mathcal{S}_B^n$ be the category whose objects are open subsets of $B\times\mathbb{C}^n$, and whose morphisms $f\colon U\to U'$ are the continuous maps $f\colon U\to U'$ such that $\pi_B\circ f =\pi_B$ (where $\pi_B$ denotes the projection $B\times\mathbb{C}^n\twoheadrightarrow B$) (and also such that $f$ is somehow locally holomorphic, but I don't really care about that bit).

It seems like some sort of slice construction, but where you take a product with $\mathbb{C}^n$, but I can never remember what such constructions are called.

Notification Bot (Jan 26 2021 at 03:39):

This topic was moved here from #general > something like a slice category? by Nathanael Arkor

Tim Hosgood (Jan 26 2021 at 03:40):

@_Notification Bot|100006 said:

This topic was moved here from #general > something like a slice category? by Nathanael Arkor

(woops, thank you!)

John Baez (Jan 26 2021 at 03:44):

So $U$ and $U'$ are spaces over $B$, meaning they'r equipped with maps to $B$, though of a very particular sort. Your equation is then saying precisely that your map $f : U \to U'$ is a map of spaces over $B$.

John Baez (Jan 26 2021 at 03:48):

So, you are describing a certain subcategory of the category of spaces over $B$.

The category of spaces over $B$ is also called the slice category of $B$, or (more descriptively yet tersely) the over category of $B$. If you look at the link you'll see a commutative triangle, and this is precisely the equation you wrote down.

John Baez (Jan 26 2021 at 03:51):

All the business about the space $U$ actually being an open subset of $B \times \mathbb{C}^n$ is too detailed and specialized to expect category theorists to have a name for it. :upside_down:

John Baez (Jan 26 2021 at 03:52):

More precisely: there might be a name for the particular subcategory of the slice category that you care about, but I've never heard of it.

Tim Hosgood (Jan 26 2021 at 03:53):

:+1:

Tim Hosgood (Jan 26 2021 at 03:55):

what about this business of taking a product (ignoring the open set stuff)? as in, this specific type of slice category where you fix some X and then look at the category of “things producted with X as objects living over X”. does this have a snappy name?

John Baez (Jan 26 2021 at 04:06):

Not that I know of. To me it's just some subcategory of the slice category. But maybe someone has made up a name for it.

John Baez (Jan 26 2021 at 04:07):

Well, I guess I'd call something like $X \times B \to B$ a "trivial bundle" in some very general sense of the word "bundle".

John Baez (Jan 26 2021 at 04:08):

So I guess I might call all the full subcategory of the slice category of $B$ consisting of objects of the form $X \times B \to B$ the "category of trivial bundles over $B$", if I had to call it something.

Tim Hosgood (Jan 26 2021 at 04:08):

it does look rather bundle-y indeed

John Baez (Jan 26 2021 at 04:09):

Yeah, a general locally trivial bundle is a bundle that's locally isomorphic to one of this "trivial" form.

John Baez (Jan 26 2021 at 04:10):

But then you're going a further step and looking at subobjects of these "trivial bundles"... but not all subobjects, just "open" ones...

Tim Hosgood (Jan 26 2021 at 04:10):

i guess i should just look up “mixed varieties” or “mixed complex spaces” up on the nlab and see if anybody has said anything there

Tim Hosgood (Jan 26 2021 at 04:11):

but my go-to is now this zulip, since you get a nice chit chat with your answers :wink:

John Baez (Jan 26 2021 at 04:11):

Just be done with it and call them Hosgood spaces.

fosco (Jan 26 2021 at 09:15):

"the zulip subcategory"

Reid Barton (Jan 26 2021 at 14:30):

I assume this is intended to get at a definition of something like a family of complex manifolds over $B$? Is this what a "mixed complex space" is?
I thought about something like this in another context recently but I didn't know if there was a name for it.

Tim Hosgood (Jan 26 2021 at 14:34):

yes, this is the definition that comes just before the definition of a mixed space (cf http://www.numdam.org/item/SHC_1960-1961__13_1_A1_0/)

Reid Barton (Jan 26 2021 at 14:35):

aha, I didn't translate my search queries into french...

Reid Barton (Jan 26 2021 at 14:41):

I'll have to read this more closely later but this looks interesting, thanks!

Tim Hosgood (Jan 26 2021 at 16:50):

(it’s next on my list of things to translate, which is why i’m trying to understand it a bit better)