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contractible; homotopic to a constant; reduced homotopy; partial simplicial object
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.
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