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There is a strange glitch on this page: the geometric realization of a cubical set (see geometric realizationealization) below) tends to have the wrong homotopy type:
That is what appears but is not a t all what the source looks like:
the geometric realization of a cubical set (see [geometric realization](#geometric realization) below) tends to have the wrong homotopy type:
What is going wrong and how can it be fixed?
Another point : does anyone know anything about symmetric cubical sets?
Strange glitch, but at the moment I am more interested in the mathematics. Could someone supply an example that the geometric realization of a cubical set tends to have the wrong homotopy type?
Off the seat of my pants: could a symmetric cubical set be a presheaf on the walking PROP that has two maps $i_0, i_1: 0 \to 1$ with a common left inverse $p: 1 \to 0$?
I do not know what you want to accomplish ? To find a notion X such that
X : cubical set
is in the "same" relation as
symmetric simplicial set : simplicial set ?
Well in the framework of skew-simplicial sets one enlarges the category of simplices (in this community also called simplex category) by a group with certain compatibility relations. This procedure can be axiomatized, there is a work of Krasauskas then of Fiodorowicz and there is also something related in the cyclic case in an unpublished manuscript of Moerdijk.
Zoran, I’m not sure who you’re addressing, but I’m going to guess it’s Tim #1.
@Zoran Marco Grandis and Luca Mauri, Cubical sets and their site, TAC, define a notion of symmetric cubical set. I met it in Goubault and Mimram o Concurrency, and it did not make too much sense to me (I like nice simple explanations -:)). I will look at the source, but it seems likely that it is a useful idea judging from what I did understand!
@Todd P. 196 (of Grandis-Mauri) gives the basic category as a PROP but I think it is more complicated than the one you suggest (??)
The example of a bad realisation is given in the n-lab entry.
Oh right, Tim, I read parts of Grandis-Mauri paper (pdf) in 2004 or so, when I was doing something with symmetric comonads (which I think Grandis defined!) in my "cyclic" preprint. So the answer to my question is presumably yes from the context I remember there, the symmetric cubical is in the sense I alluded to. So you add transpositions to the generators of the cubes category just like Connes added transposition to the generators of simplex category to get cycle category, but such that the elementary transposition square equals identity. This way one does some sort of a (wreath? crossed?) product of symmetric group with the cube category, namely one adds commutation relation to have some sort of a skew-product structure, just like in Krasauskas general work on skew-simplicial groups. As symmetric case is the biggest allowing that structure all other skew-cubical groups correspond to taking subgroups of teh symmetric group so that the commutation relations are closed. This was completely classified in the simplicial case, I do not know if the analogous result is published in the cubical case.
Is this timely or what?
a preprint: A symmetric cubical category associated to a directed space
Yes!
I just saw the Cat-list notice 20 minutes ago.
I have to admit that I still find it hard to see where constructions such as this are headed.
It seems un-surprising that we can give some definition of a higher fundamental category of a directed space, even if depending on the model used checking the consistency of the definition may be tedious. So with every definition proposed, there should go some theorem saying something about this definition that shows us that the definition does serve to capture some phenomenon that we wish to capture. Do we have such theorems here? Or do we have at least desirerata spelled out, for what kinds of theorems we expect the definition to be necessary for?
If there are good answers to this, then we should put them into the $n$Lab in the thread “motivations for directed homotopy” that Tim started. I would be very interested in learning more in this direction.
@Urs I think some of the answers are in the Goubault-Mimram paper that I have linked to. There are well established models of concurrency coming from Computer Science, and in the more adventurous ones there is a real interest in probabilistic non-determinism, with practical applications not only to ’languages’, but to models for quantum space-time, and operational research applications via Petri nets, discrete event systems, etc. (leading to tropical monoids etc.). The symmetric cubical set structure is used to define higher dimensional automata (Vaughan Pratt, etc. but here in the paper by Goubault and Mimram). They show how there are neat adjunctions between HDAs and more or less all the other models for concurrency, (at least the ones that I mentioned). They comment that the Mazurkiewicz trace monoid is related to the fundamental category construction as developed by Goubault, Raussen, Fajstrup etc. but the interesting idea is that things like Petri nets have other aspects that tie them into rewriting, project planning, and so on.( In his book, Grandis talks about Weighted homotopy in which the Lawvere generalised metric space idea is used and this does seem related.)
My feeling is there should be a link here somewhere to Tom Leinster’s work, but that is just a feeling at present.
There is a strong link to languages used for describing ’systems’ both in engineering and software design. My paln for the concurrency entries is to get the definitions down, then go back and describe (where possible) the relations with cubical sets, and then to develop the directed homotopy of each. Sometimes it will say very little, but in some of the situations (where there are for instance, three machines bit only two can be operated at any given time,) there will be higher homotopy information around. That is vague and unclear to me, so I am trying to work this out fairly systematically in the entries as I create them.
I need a store of example situations that correspond to the various aspects these things model as well!
It is shameful to admit, but the typesetting of Grandis’ papers makes them hard for me to read. Why not use LaTeX? This looks like he’s using Word :|
I know. Shameful, but true…
I think he may do.
Hey Tim,
Not that you’ve got tons of free time and neither do I, but it would be fun to work on some “kindergarten” version of Grandis’ stuff on finite topological spaces (which we both know are not as boring as they may seem at first).
Want to start a mini project together? I’d like to go through Grandis’ stuff and finitize it if possible.
By the way, how is the typesetting in his book? Is it more pleasant to read than the papers?
His book looks to be in Latex. I found it hard going as he makes few concessions to the reader (e.g. the examples are concentrated in certain places whilst I would prefer an example taken apart each time a new type of directed homotopy is discussed. He spends a lot of pages looking at different strengths of cylinder functor. This is useful but quite dry (for me). The book is well written.
My thoughts on finite spaces are that there should be a small development of the ordinary homotopy theory of finite spaces (more than at present, e.g. lots more examples !) and we would have to look at the fact that finite posets make up a class of them. We could try the first part slowly if you like. Nice idea.
My thoughts on finite spaces are that there should be a small development of the ordinary homotopy theory of finite spaces (more than at present, e.g. lots more examples !) and we would have to look at the fact that finite posets make up a class of them. We could try the first part slowly if you like. Nice idea.
That sounds cool and I’m all for doing things slowly :)
Should we put some material directly to the nLab or start on one of our personal webs (maybe your’s since mine contains a bunch of half-baked notes) and transfer later?
If we start with some motivation Why study finite spaces and develop that in the n-Lab, we can then pause, see what direction we want to go, do some work in my personal web and when we are clear on the structure that fits then transfer.
@Eric Have a look at http://ncatlab.org/timporter/show/HomePage near the bottom of the page.
I thought there was an n-Lab page on finite topological spaces already, but cannot find one!
@Eric Have a look at http://ncatlab.org/timporter/show/HomePage near the bottom of the page.
Cool :)
By the way, you can link to personal web pages similar to nLab pages using markdown, but you need to specify the web, e.g.
Check the “Source”. It may come in handy if we work on some things there.
A few minutes back I wanted to point somebody to the fact that the cube category is a test category but only a strict test category when refined to cubes with “cube connection”.
I found the information on this somewhat unpleasantly scattered over $n$Lab entries. I have now briefly edited various entries with “cube” in the title in an attempt to improve on the situation. But more needs to be done.
(If you are a cube afficionado and wish to see cubes play a more prominent role in current homotopy theory, maybe you want to consider improving on the relevant $n$Lab entries…)
The thing that is ’wrong’ is that under geometric realization of ordinary cubical sets a cartesian product of cubical sets is not taken to the cartesian product of geometric realization of each factors, not even up to weak equivalence, in general.
I’m glad you explained this, Urs, because I couldn’t tell how Tim #6 answered my #2 (which nLab entry? where?).
I’m also glad you put ’wrong’ in quotes, because it’s obviously ’right’ in another sense (no, cubical realization doesn’t preserve cartesian products, but it does take the Day convolution on cubical sets to cartesian products, quite by design).
Hi Todd,
I suppose in this thread “the entry” may be taken to be a bound variable with value cubical set. Specifically the section Cubical sets in homotopy theory mentions what I just said.
I think it is easy to see what’s going on in full detail by just looking at the case of the product of the standard interval with itself (as mentioned there).
In simplicial sets, one finds that the product simplicial sets $\Delta[1] \times \Delta[1]$ is indeed a square, because some degenerate 2-cells in $\Delta[1]$ become parts non-degenerate pairs of cells in the product. “The two non-degenerate triangles in $\Delta[1] \times \Delta[1]$ originate in degenerate triangles in $\Delta[1]$”, if you wish.
That the analogous statement fails for $\Box[1] \times \Box[1]$ is due to lack of just such degenerate higher cells in $\Box[1]$. This is what the passage to cubical sets “with connection” improves on: the “connections” (unfortunate term, I think) are precisely the extra degenerate cells that need to be there in order to give rise to homotopy-good cartesian products.
I had actually looked at cubical set, but clicked immediately to the subsection on geometric realization. I hadn’t noticed there was a longish preamble before that subsection which contains the discussion. Anyway, the point is clear.
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