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    • CommentRowNumber1.
    • CommentAuthorEric
    • CommentTimeOct 18th 2010

    I was just sitting down to begin reading

    and came across this line

    We will show that Whitehead’s Theorem does not hold for finite spaces: there are weak homotopy equivalent finite spaces with different homotopy types. This distinction between weak homotopy types and homotopy types is lost when we look into the associated polyhedra (because of Whitehead’s Theorem) and, in fact, the essence of Quillen’s conjecture lies precisely in the distinction between weak homotopy types and homotopy types of finite spaces.

    Is this obvious to you guys? It is over my head, but it sounded like maybe someone here would find the statement interesting. Sorry if it is completely trivial/uninteresting.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeOct 18th 2010

    Any T_0 space ’is’ a poset so the ’weak homotopy type it represents is a bit like that of the nerve of that poset. Now look at the dual poset. The two nerves will have homeomorphic realisations so …. but there may be very few maps from XX to X opX^{op}, too few to get their homotopy types to be the same. I think that is the reason for what he says.

    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeOct 19th 2010

    On page 20, I found this bit of trivia interesting:

    Given a finite space XX, there exists a homotopy equivalent finite space X 0X_0 which is T 0T_0. That means that for any two points of X 0X_0 there exists an open set which contains only one of them. Therefore, when studying homotopy types of finite spaces, we can restrict our attention to T 0T_0-spaces.

    I don’t know enough about homotopy to appreciate whether all I care about is homotopy equivalence though. My motivation is ultimately about modeling physics, dynamic systems, etc.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeOct 19th 2010

    @Eric Do you want more on that in the project pages. I was taking the stuff I put there from a Dagstuhl -TCS paper which looked at determining finite spaces from data as a T 0T_0-quotient space. What I did not consider was any idea of applying things say to a ’causal space’ .The construction I used was in Sorkin’s work (before he found the same models gave causets), but that does not really make the transition to evolving spaces in any way. Suppose one has a pospace say based on cubes as in the stuff on concurrency that was at the start of the directed space idea. My impression is that if one looked at directed paths through such a space then at most times there would be a ’locally stable time-slice’. My thought is most easily understood by the usual picture of the torus (on end) with height giving a notion of time, The time slices are then circles etc as one expects from Morse theory. Now perturb the height function a bit within the space of Morse functions, the time slice evolution will not change much at least intuitively. (Sort of persistent homology seems relevant here.) Now start modelling things with finite spaces. Observe the torus using an open cover and time via an open cover as well. I have not thought about what happens to the time slices.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 20th 2010

    I was thinking re: perturbing time measurements, that one could have, as in persistent homology, a variable ’radius’ for the time direction. Thus instead of homology groups indexed by \mathbb{R}, they are indexed by 2\mathbb{R}^2. Assume our set of observations sits in some ambient n+1\mathbb{R}^{n+1}. Over a point (ε,d)(\varepsilon,d) one takes cylinders [tε,t+ε]×B(x,d)[t-\varepsilon,t+\varepsilon]\times B(x,d) centred on the observation at (t,x)(t,x) (or possibly ellipsoids with one axis ε\varepsilon and the nn others just dd). Then one forms the simplicial complex as per usual by taking edges etc. given by the incidence of cylinders and takes the homology of this.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeOct 20th 2010

    Nice idea. I had thought something like that but never developed it. I seem to remember Carlsson looking at something a bit like that as well. (But if I start on persistent homology I will never finish all the other topics that I have been foolish enough to start. :-)) I am trying to link n-POV and all that sort of stuff. I am convinced it is worth doing. At the moment this involves taking bits of theory (thought of as jigsaw puzzle pieces) and turning them face up on the theory table and see if someone can find bits of the giraffe’s neck that fit with the bit I have near me!

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeOct 20th 2010

    That should have been ’all this sort of stuff’ meaning finite spaces, modal logics, HDAs. (The forum is very slow at the moment.)