Timeline for What properties make $[0,1]$ a good candidate for defining fundamental groups?
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| May 7, 2012 at 10:08 | comment | added | Ronnie Brown | I'd like to draw attention to the paper: Dedecker, Paul and Valderrama, Jerko, Graphes et cographes sur une cat\'egorie abstraite. Application `a l'homotopie, C. R. Acad. Sci. Paris S\'er. A-B,262, 1966, A377--A380, as an early discussion of an interval object. | |
| Mar 26, 2012 at 5:18 | comment | added | Mike Shulman | It's not true that the two-point set is completely useless for measuring homotopy. "Homotopies" using the Sierpinski space (the non-discrete, non-codiscrete topology on the two-point space) detect exactly the specialization ordering. And there is a map from [0,1] to the Sierpinski space, so any specialization inequality induces a path in the traditional sense. For the "usual" sorts of spaces, the specialization ordering is boring, but for (e.g.) finite topological spaces, it contains all the homotopy information. | |
| Mar 26, 2012 at 3:20 | comment | added | David Roberts♦ | Pullback in the opposite category ^_^ No, I just had a brain explosion - fixed. | |
| Mar 26, 2012 at 3:19 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Mar 26, 2012 at 2:05 | comment | added | Todd Trimble | I find the notation $[0, 1] \times_{0, 1} [0, 1]$ confusing, since it suggests a pullback; I expect what you mean is a pushout which glues the endpoint 1 of the first interval to the endpoint 0 of the second. On a different note, I think the linear orderedness might be a slight red herring, e.g., the unit interval in the Dedekind reals in any Grothendieck topos is the terminal coalgebra for the sort of double-gluing construction outlined in the first sentence of this comment (if done constructively correctly; see the Elephant), as a functor on bipointed objects. | |
| Mar 26, 2012 at 1:40 | comment | added | David Roberts♦ | You need to be able to define 'contractible' somehow in order to make a lot of this machinery to work. In order to do that, you need some sort of model category (not necessarily a Quillen one) or some sort of category with a cylinder object. The unit interval can be defined as a terminal coalgebra of an endofunctor of the category of intervals, see the examples section at ncatlab.org/nlab/show/terminal+coalgebra, but this is not a homotopy-theoretic construction. Note also that it assumes linearly ordered, not merely bipointed as you have done. | |
| Mar 26, 2012 at 1:26 | comment | added | Daniel Miller | Nice answer, but I don't feel that you are getting to the heart of my question. Showing that the unit interval behaves well "up to homotopy" is presupposing the fact that we have defined homotopy using the unit interval. If we, so to speak, we were not already using $[0,1]$ to do homotopy theory, then how could we identify the relevant categorical properties of $[0,1]$? By categorical here, I mean expressible in the category of topological spaces and continuous maps (not up to homotopy). | |
| Mar 26, 2012 at 1:06 | history | answered | David Roberts♦ | CC BY-SA 3.0 |