Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorEric
   • CommentTimeOct 18th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexI was just sitting down to begin reading * Jonathan Barmak, [Topología Algebraica de Espacios Topológicos Finitos y Aplicaciones](http://www.math.kth.se/~jbarmak/tesisfinal2.pdf) 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.

   I was just sitting down to begin reading

   • Jonathan Barmak, Topología Algebraica de Espacios Topológicos Finitos y Aplicaciones

   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.

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeOct 18th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexAny 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 $X$ to $X^{op}$, too few to get their homotopy types to be the same. I think that is the reason for what he says.

   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 $X$ to $X^{op}$, too few to get their homotopy types to be the same. I think that is the reason for what he says.

3. • CommentRowNumber3.
   • CommentAuthorEric
   • CommentTimeOct 19th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexOn [page 20](http://www.math.kth.se/~jbarmak/tesisfinal2.pdf#Page=20), I found this bit of trivia interesting: >Given a finite space $X$, there exists a homotopy equivalent finite space $X_0$ which is $T_0$. That means that for any two points of $X_0$ 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 $T_0$-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.

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

   Given a finite space $X$, there exists a homotopy equivalent finite space $X_0$ which is $T_0$. That means that for any two points of $X_0$ 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 $T_0$-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.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeOct 19th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItex@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 $T_0$-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.

   @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 $T_0$-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.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 20th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI 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 $\mathbb{R}^2$. Assume our set of observations sits in some ambient $\mathbb{R}^{n+1}$. Over a point $(\varepsilon,d)$ one takes cylinders $[t-\varepsilon,t+\varepsilon]\times B(x,d)$ centred on the observation at $(t,x)$ (or possibly ellipsoids with one axis $\varepsilon$ and the $n$ others just $d$). 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.

   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 $\mathbb{R}^2$. Assume our set of observations sits in some ambient $\mathbb{R}^{n+1}$. Over a point $(\varepsilon,d)$ one takes cylinders $[t-\varepsilon,t+\varepsilon]\times B(x,d)$ centred on the observation at $(t,x)$ (or possibly ellipsoids with one axis $\varepsilon$ and the $n$ others just $d$). 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.

6. • CommentRowNumber6.
   • CommentAuthorTim_Porter
   • CommentTimeOct 20th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexNice 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!

   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!

7. • CommentRowNumber7.
   • CommentAuthorTim_Porter
   • CommentTimeOct 20th 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThat should have been 'all this sort of stuff' meaning finite spaces, modal logics, HDAs. (The forum is very slow at the moment.)

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