$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top}
\DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such that the singular simplicial complex functor $\Sing\_\bullet:\Top \to \sSets$ is fully faithful ?



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> Is $\Hom\_{\Top}(X,Y)=\Hom\_{\sSets}(\Sing\_\bullet X, \Sing\_\bullet Y)$ for $X$ and $Y$ topological spaces nice enough ?
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Is there a good reference for facts like this ?


Note that $X:=\Bbb{Q}$ and $Y:=\Bbb Z$ form a counterexample, as the singular simplicial complex of a totally disconnected space is constant. Thus totally disconnected spaces are not nice enough. See details at [Is the singular simplicial functor full](https://mathoverflow.net/questions/261925/is-the-singular-simplicial-functor-full).



Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?
Here is a simple counterexample with $X = Y = \mathbb{R}$:
Send a simplex $\sigma : |\Delta^n| \to \mathbb{R}$ to the affine function $F(\sigma) : |\Delta^n| \to \mathbb{R}$ with the same values at the vertices of $|\Delta^n|$. This is the identity on 0-simplices, so it could only come from the identity map of $\mathbb{R}$, but it's not the identity for $n > 0$.


Your question is implicitly about the counit $|\mathrm{Sing}\,X| \to X$. In general, $|\mathrm{Sing}\,X|$ is much bigger than $X$. The "problem" is that there are not enough maps in the simplex category (i.e. $|{-}| : \Delta \to \mathrm{Top}$ is not fully faithful). That is why, above, the value of $F(\sigma)$ at a non-vertex point $x \in |\Delta^n|$ did not have to have any particular relation to the value of $\sigma$ at $x$.


