Timeline for Degeneracies for semi-simplicial Kan complexes
Current License: CC BY-SA 2.5
14 events
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| Dec 3, 2012 at 3:32 | comment | added | Peter May |
There is a nice quick description of the left adjoint to the forgetful functor from simplicial sets to semi-simplicial sets in Fritsch and Piccinini Cellular structures in algebraic topology'' together some not so usual material. See Section 4.4. They use the name presimplicial sets'' for these animals. They are related to categories without identity morphisms, which appear occasionally in the combinatorial literature.
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| Mar 8, 2011 at 14:30 | vote | accept | Mike Shulman | ||
| Mar 8, 2011 at 12:23 | comment | added | Buschi Sergio | I find in wiky: en.wikipedia.org/wiki/Delta_set Please, can someone send me the article: Rourke, C. P.; Sanderson, B. J. (1971). "Δ-Sets I: Homotopy Theory". I wish like read about...([email protected]) Thank anyway | |
| Mar 8, 2011 at 11:53 | history | edited | John Klein | CC BY-SA 2.5 |
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| Mar 7, 2011 at 19:51 | comment | added | John Klein | No, the outline that I gave does not show how to fill in the degeneracies to obtain the simplicial set. Since they are working hard to get the result, it is likely the case that there is no functorial way to fill in the degeneracies. | |
| Mar 7, 2011 at 18:33 | comment | added | Mike Shulman | The paper does indeed show that $X\to j^* j_! X$ is an equivalence, in the sense that it induces an equivalence upon geometric realization. If I'm not mistaken, this is a particular case of the theorem that the "fat geometric realization" of a good simplicial space is equivalent to its ordinary realization. You seem to think it would follow from this that X can be given degeneracies, but I don't see that immediately, can you explain? The paper takes 3 more sections to get to a proof of that fact, which I haven't digested yet; they get it as a corollary of a simplicial approximation theorem. | |
| Mar 7, 2011 at 18:11 | comment | added | John Klein | Kan refers to what we now call "simplicial sets" as c.s.s. complexes. I guess "c.s.s." refers to the Eilenberg-Zilber "complete semi-simplicial." | |
| Mar 7, 2011 at 18:05 | comment | added | Mike Shulman | Also, as far as I can tell "semi-simplicial complex" was originally used by Eilenberg and Zilber for the version without degeneracies; they added the adjective "complete" when there the degeneracies were given. Later people seem to have dropped the "complete" in referring to the version with degeneracies, and then later dropped the "semi-". So calling the degeneracy-less ones "semi-simplicial" is actually the more faithful to the original! (-: | |
| Mar 7, 2011 at 18:00 | comment | added | Mike Shulman | Thanks! I find "semi-simplicial set" more evocative, since "semi-" is used for "without identities" in some other contexts. I'll look up the paper. | |
| Mar 7, 2011 at 13:31 | comment | added | John Klein | I like the term "simplicial set w/o degeneracies," even though its long-winded. | |
| Mar 7, 2011 at 13:26 | comment | added | Tyler Lawson | There have been a few people who have explicitly adopted the term "semi-simplicial set" for this type of object in recent years; McClure is one. | |
| Mar 7, 2011 at 13:25 | history | edited | John Klein | CC BY-SA 2.5 |
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| Mar 7, 2011 at 12:58 | history | edited | John Klein | CC BY-SA 2.5 |
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| Mar 7, 2011 at 12:41 | history | answered | John Klein | CC BY-SA 2.5 |