Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set).

Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor.

For every $X$ we have a natural map that is a homotopy equivalence $\epsilon_X: ∣Sing(X)∣\rightarrow X$.

Is there a natural homotopy inverse? Said differently, is there a natural transformation $\text {Id} \Rightarrow∣Sing(-)∣$ consisting of homotopy inverses to the $\epsilon_X$’s?

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    $\begingroup$ No, use naturality for the inclusions of points into a space $X$. $\endgroup$ – Oscar Randal-Williams Feb 11 at 23:43
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    $\begingroup$ Could you write with more detail? $\endgroup$ – Lao-tzu Feb 13 at 22:13
  • $\begingroup$ Yes, I too would appreciate more details! $\endgroup$ – safety stegosaurus Feb 14 at 22:47

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