Timeline for A canonical and categorical construction for geometric realization
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Jun 17, 2010 at 16:03 | comment | added | David Carchedi | P.S., I've edited the question to include a possible lead. | |
| Jun 17, 2010 at 14:29 | comment | added | Oscar Randal-Williams | @Tyler: So it is. Presumably this has to do with the fact that face inclusions of these "finite simplices" are not cofibrations? | |
| Jun 17, 2010 at 13:52 | comment | added | Tyler Lawson | @David: Your concerns are valid; e.g. the standard simplicial circle (having only two nondegenerate simplices) has realization, using the finite spaces mentioned by Oscar Randall-Williams, as the Sierpinski two-point space, which is contractible. | |
| Jun 17, 2010 at 11:33 | comment | added | Steven Gubkin | Sure, these are all very valid concerns. I have just always felt that bringing the continuum into the picture seems kind of disingenuous when we are only (usually) thinking about a finite number of relationships like "this is a face of this which is glued to this". So I hope that this will work out to have all the nice properties we want it to have for philosophical reasons - just by recording these relations between simplicies in a convenient finite space, everything we care about is preserved. I will try and flesh out these ideas more today. | |
| Jun 17, 2010 at 9:42 | comment | added | David Carchedi | The reason is, the geometric realization functor has two important properties: 1.) It is conservative, 2.) it is left-exact. Especially the later is used to relate simplicial categories with topological categories, etc. Moreover, it seems that since the geometrical realization enjoys these properties, that it is "more correct" than any homotopy equivalent replacement. | |
| Jun 17, 2010 at 9:40 | comment | added | David Carchedi | This does seem very interesting, but, I have some concerns. If we could definte a "natural" functor $\Phi$ from $\Delta$ to $FinTop$ (finite topological spaces), just because for each $[n]$ $\Phi([n])$ is weakly equivalent to $\Delta^{n}$, it doesn't seem obvious to me that its left Kan-extension (denote it also by $\Phi$) would have $\Phi(X)$ w.e. to $|X|$ for each simplicial set $X$. In a more fundamental way, I am against replacing $|X|$ by something which is homotopy equivalent and saying it's "good enough" for algebraic topology. (will continue in another comment). | |
| Jun 17, 2010 at 0:28 | comment | added | Dan Ramras | Since there aren't any references for finite spaces on Wikipedia, I'll point out that Peter May's website contains some nice notes on the topic, and there are also recent papers by Minian and Barmak on the arXiv. The original source for the result (stated in the question) about weak homotopy equivalence is McCord's paper (Duke J., 1966). Richard Stong also wrote some early papers on the topic. | |
| Jun 16, 2010 at 18:24 | comment | added | Steven Gubkin | Yes, that is the first idea I had, and what I am playing around with. The problem that I am having is figuring out how to glue these guys appropriately: How can I recover (a finite substitute for) a circle, or a torus? | |
| Jun 16, 2010 at 18:17 | comment | added | Oscar Randal-Williams | I suspect the finite space you want to use as a replacement for $\Delta^n$ is the one with elements the simplices of $\Delta^n$, and topology given by the preorder of face inclusions i.e. so the faces are the closed subsets. | |
| Jun 16, 2010 at 18:14 | history | edited | Steven Gubkin | CC BY-SA 2.5 |
deleted 12 characters in body
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| Jun 16, 2010 at 17:13 | history | answered | Steven Gubkin | CC BY-SA 2.5 |