Geodesic
Contractible simplicial objects
Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 4, pp. 473-495 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.14712/1213-7243.2019.023
Classification : 18G30, 55U10
Keywords: contractible; homotopic to a constant; reduced homotopy; partial simplicial object 
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Barr, Michael; Kennison, John F.; Raphael, Robert. Contractible simplicial objects. Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 4, pp. 473-495. doi: 10.14712/1213-7243.2019.023

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