Michael Barr, John F. Kennison, Robert Raphael
Contractible simplicial objects

Comment.Math.Univ.Carolin. 60,4 (2019) 473-495.

Abstract: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.

Keywords: contractible; homotopic to a constant; reduced homotopy; partial simplicial object

DOI: DOI 10.14712/1213-7243.2019.023
AMS Subject Classification: 18G30 55U10