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I put some very very basic notes down at Finite Topological Spaces-Project (timporter) trying to get the ball rolling.
I looked out a paper the other day and will start discussing that. (It was one of those from Gabriel Minian’s student.)
I added some references (since I couldn’t remember which student you were referring to) :)
I took some stuff from a paper I wrote and have put it there. It is about the way that finite observations lead to finite $T_0$-spaces.
The problem is that the partial order is very much like the face order on the faces of a simplicial complex. (We could be looking for finite biposets as one of the models for directed finite spaces! Those beasties have occurred in physics applications.)
I mentioned this project by the way over at the Café, in a comment here.
While you are working on this project on the personal web, I suppose we should collect the standard facts in an $n$Lab entry finite topological space.
@Urs. By all means. As the ’project’ wanders around looking for nice ideas on finite spaces, and nice links, there will be informal stuff appearing there, some of which may be useful in the main Lab. (For instance, using McCord’s result how might a finite model of a weak homotopy type reflect properties of the other models. A typical result is one by Jonathan Barmak : Any space with the same homotopy groups as $S^n$ has at least 2n+2 points. )
But Eric is interested in seeing what might give directed finite spaces in some sense. We will see.
Question: do analogues of finite topological spaces exist in other contexts? e.g. in other toposes than Set. Probably they are just poset objects?? (This may be a daft question, …!)
(Update) I have started giving the definition of Christensen-Crane’s notion of causal site (timporter). ( I thought this was discussed on the Café at some point but could not find it.) Does anyone know of any further work on that definition as in the form it is given here it looks as if it needed ’simmering over a low flame’ for a few more hours before it was tender enough for me to understand! In other words, I do not quite see where it is going from my own point of view.
As I said elsewhere, as a finite topological space (if $T_0$) is just a finite poset, then perhaps we need to look at some sort of double poset to get a directed finite space. Thoughts welcome. Crane and Christensen do look at the bisimplicial nerve of their structure later in the paper, but I have not yet got there in my discussion!
Hi Tim,
you had a “(something wrong with the formatting)”-remark on your page. I have fixed it: the bullets of the bullet-item list must not be indented with respect to the left margin of the context they are sitting in. I moved them two spaces to the left to make the output come out right. Have a look to see what I mean.
@Urs Thanks. I had them on the left before and the numbering went wrong They look good now.
Added example to causal site (timporter). Their motivating example looks strong to me. (Rick Blute and various others also found this when looking at the paper. It means that a diamond needs to have no intersection with another one if is is to be $\lt$ it. Rick suggested looking at power domains as a way around the difficulty.)
Thinking about the idea of directed finite spaces from this POV and looking at the ’observational’ reasons for using finite spaces = posets, there might be good reasons for viewing power domain type constructions as a useful source of techniques as they (if I understand correctly) look at the poset structures on the set of subsets of a poset. (Ok with special properties on the poset.)
Quick thoughts after a long day…
Simplices are the “shape of choice” for studying spaces.
I suspect that as useful as simplices are for studying spaces, diamonds are the “shape of choice” for directed spaces.
A directed space has a notion of time. In some cases you can collapse a directed space along this time direction to get an undirected space, i.e. (just) a space.
If you collapse a diamond, you get a bunch of simplices. This is illustrated at diamonation (ericforgy).
The reverse of collapsing, i.e. extrusion, is also interesting.
You can take a triangulated finite space and extrude it into a directed finite space in such a way that the simplices extrude into diamonds.
You assume non-branching time in some of that!
The notion of time is local I think. A global time depends on a synchronisation of different bits and that can be problematic. a directed space (whatever that may mean) seems to have the ability to say x happened before y for some pairs x and y, but not necessarily all pairs.
I do not think diamonds are the be-all and end-all in this game. The point of cubes and simplices is really the relationships between the whole thing and its faces and as yet I do not see that aspect for diamonds.
What do you mean by triangulated finite space?
You assume non-branching time in some of that!
Not really because I said “In some cases” :)
The notion of time is local I think.
Yeah, I agree.
A global time depends on a synchronisation of different bits and that can be problematic.
Yeah, and I would never want to impose a global time coordinate.
a directed space (whatever that may mean) seems to have the ability to say x happened before y for some pairs x and y, but not necessarily all pairs.
Yeah, just like Lorentzian spacetime :)
Somewhere, I made a comment about this. The proportion of points that are uncomparable relates to the “speed of light” (or maybe better “speed of information”) of the directed space, i.e. the maximum local speed (which need not be constant). If all points are comparable, assuming no obstructions, then the speed of light is infinite. For example, Gallilean space is totally ordered. Minkowski space is partially ordered.
I do not think diamonds are the be-all and end-all in this game. The point of cubes and simplices is really the relationships between the whole thing and its faces and as yet I do not see that aspect for diamonds.
Well, in a finite space, a cube is a special case (the defining case) of a diamond. A cube is not a diamond in a continuum space though.
What do you mean by triangulated finite space?
Mangled. I meant something like “A finite space which is a triangulation of some continuum space”. Maybe “finite triangulation”.
For me, triangulations are really open covers in disguise. SO take a finite open cover /finite set of observations and that seems better to me. Must rush now.
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