# Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

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?

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