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For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.
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The classifying space of an infinite totally ordered set is contractible
I am not sure if you consider this explicit, but here you go:
Choose a point $x_0\in X$, and let $F\colon X\to X$ be the functor that sends $x$ to itself if $x\geq x_0$, and to $x_0$ otherwise. There …