| Title:
|
Contractible simplicial objects (English) |
| Author:
|
Barr, Michael |
| Author:
|
Kennison, John F. |
| Author:
|
Raphael, Robert |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
473-495 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We raise the question of when a simplicial object in a catetgory is deemed contractible. The literature offers three definitions. One is the existence of an ``extra degeneracy'', indexed by $-1$, which does not quite live up to the name. This can be strengthened to a ``strong extra degeneracy". Another possibility is that it be homotopic to a constant simplicial object. Despite claims in the literature to the contrary, we show that all three are distinct concepts with strong extra degeneracy implies extra degeneracy implies homotopic to a constant and give explicit examples to show the converses fail. (English) |
| Keyword:
|
contractible |
| Keyword:
|
homotopic to a constant |
| Keyword:
|
reduced homotopy |
| Keyword:
|
partial simplicial object |
| MSC:
|
18G30 |
| MSC:
|
55U10 |
| idZBL:
|
Zbl 07177884 |
| idMR:
|
MR4061357 |
| DOI:
|
10.14712/1213-7243.2019.023 |
| . |
| Date available:
|
2020-02-10T16:48:49Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147968 |
| . |
| Reference:
|
[1] Barr M.: Acyclic Models.ACRM Monograph Series, 17, American Mathematical Society, Providence, 2002. MR 1909353, 10.1090/crmm/017/05 |
| Reference:
|
[2] Beck J. M.: Triples, algebras and cohomology.Repr. Theory Appl. Categ. 2 (2003), 1–59. Zbl 1022.18004, MR 1987896 |
| Reference:
|
[3] Goerss P. G., Jardine J. F.: Simplicial Homotopy Theory.Progress in Mathematics, 174, Birkhäuser, Basel, 1999. MR 1711612 |
| Reference:
|
[4] González B. R.: Simplicial Descent Categories.Translated and revised version of 2007 Thesis from University of Seville, available at arXiv:0804.2154v1 [math.AG], 2008. MR 2864852 |
| Reference:
|
[5] Hamada S.: Contractibility of the digital $n$-space.Appl. Gen. Topol. 16 (2015), no. 1, 15–17. MR 3338816, 10.4995/agt.2015.1826 |
| Reference:
|
[6] Khalimskiĭ E. D.: The topologies of generalized segments.Dokl. Akad. Nauk SSSR 189 (1969), 740–743 (Russian). MR 0256359 |
| Reference:
|
[7] Linton F. E. J.: Applied functorial semantics, II.Sem. on Triples and Categorical Homology Theory, ETH, Zürich, 1966/67, Springer, Berlin, 1969, pages 53–74. MR 0249485 |
| Reference:
|
[8] Meyer J.-P.: Bar and cobar constructions. I.J. Pure Appl. Algebra 33 (1984), no. 2, 163–207. MR 0754954, 10.1016/0022-4049(84)90005-7 |
| Reference:
|
[9] Riehl E.: Categorical Homotopy Theory.New Mathematical Monographs, 24, Cambridge University Press, Cambridge, 2014. MR 3221774 |
| . |
