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Keywords:
contractible; homotopic to a constant; reduced homotopy; partial simplicial object
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.
References:
[1] Barr M.: Acyclic Models. ACRM Monograph Series, 17, American Mathematical Society, Providence, 2002. DOI 10.1090/crmm/017/05 | MR 1909353
[2] Beck J. M.: Triples, algebras and cohomology. Repr. Theory Appl. Categ. 2 (2003), 1–59. MR 1987896 | Zbl 1022.18004
[3] Goerss P. G., Jardine J. F.: Simplicial Homotopy Theory. Progress in Mathematics, 174, Birkhäuser, Basel, 1999. MR 1711612
[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
[5] Hamada S.: Contractibility of the digital $n$-space. Appl. Gen. Topol. 16 (2015), no. 1, 15–17. DOI 10.4995/agt.2015.1826 | MR 3338816
[6] Khalimskiĭ E. D.: The topologies of generalized segments. Dokl. Akad. Nauk SSSR 189 (1969), 740–743 (Russian). MR 0256359
[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
[8] Meyer J.-P.: Bar and cobar constructions. I. J. Pure Appl. Algebra 33 (1984), no. 2, 163–207. DOI 10.1016/0022-4049(84)90005-7 | MR 0754954
[9] Riehl E.: Categorical Homotopy Theory. New Mathematical Monographs, 24, Cambridge University Press, Cambridge, 2014. MR 3221774
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