5
$\begingroup$

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

Is $\Hom_{\Top}(X,Y)=\Hom_{\sSets}(\Sing_\bullet X, \Sing_\bullet Y)$ for $X$ and $Y$ topological spaces nice enough ?

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.

Improve this question
$\endgroup$
3
  • 9
    $\begingroup$ This will almost never be the case. For example it fails even for $X = Y = \mathbb{R}$. What class of "nice" spaces excludes the real line? $\endgroup$
    – Chris Schommer-Pries
    Commented Mar 19, 2023 at 18:28
  • $\begingroup$ @ChrisSchommer-Pries: Is this easy to see, and does it also fail for $X=Y=[0,1]$ ? What is an example of a map $Hom_{sSets}(Sing_\bullet\Bbb{R},Sing_\bullet\Bbb{R})$ which does not come from a continuous map ? $\endgroup$
    – user494312
    Commented Mar 19, 2023 at 19:56
  • $\begingroup$ It is probably enough to explain where would such a map send a path $\gamma:[0,1]\to \Bbb{R}$. $\endgroup$
    – user494312
    Commented Mar 19, 2023 at 20:01

1 Answer 1

Reset to default
15
$\begingroup$

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 $x$.

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ The failure (though not the counterexample) is also mentioned in a comment of @ChrisSchommer-Pries. $\endgroup$
    – LSpice
    Commented Mar 20, 2023 at 14:46
  • $\begingroup$ Thank you. Though, is there a positive result saying that any counterexample is of this kind ? Say, comes from a natural transformation $|-|\implies|-|$ or something like that (actually, I do not think that your conterexample does arise in this way, so what I suggest is not enough.). In other words, to make formal your remark "Your question is implicitly about the counit $|SingX| \to X$." $\endgroup$
    – user494312
    Commented Mar 20, 2023 at 15:05
  • 1
    $\begingroup$ What I meant is that, for fixed X, your equation holds for all Y if and only if the counit |Sing X| -> X is a homeomorphism. And both |Sing X| and (by assumption) X are "nice" spaces, so if (as usual) they are different, we should be able to see the difference using maps into another "nice" space Y. $\endgroup$
    – Reid Barton
    Commented Mar 21, 2023 at 4:38

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .