Let $(X,x\_0)$ be a pointed simplicial set. Assume if you like that $X$ is the nerve of a category but *do not* assume that $X$ is a Kan complex.


Because $Ex^\infty X$ is a Kan complex, every homotopy class $\alpha \in \pi\_n(X,x\_0)$ may be represented by a map $sd^k \Delta[n] \to X$ such that the restriction $sd^k \partial \Delta[n] \to X$ is constant at $x\_0$. I'm wondering about different "normal forms" for homotopy classes.


For instance, consider subidivided cubes. In dimension 2, I think they should look like this:


$\require{AMScd}
D^2\_0 =
\begin{CD}
\bullet
\end{CD} \\
D^2\_1 = 
\begin{CD}
\bullet @>>> \bullet @<<< \bullet \\
@VVV @VVV @VVV \\
\bullet @>>> \bullet @<<< \bullet \\
@AAA @AAA @AAA \\
\bullet @>>> \bullet @<<< \bullet \\
\end{CD} \\
D^2\_2 = 
\begin{CD}
\bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\
@VVV @VVV @VVV @VVV @VVV \\
\bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\
@AAA @AAA @AAA @AAA @AAA \\
\bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\
@VVV @VVV @VVV @VVV @VVV \\
\bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet\\
@AAA @AAA @AAA @AAA @AAA \\
\bullet @>>> \bullet @<<< \bullet @>>> \bullet @<<< \bullet
\end{CD} \\
D^2\_3 = \dots
$


**Questions:**


1. Can every $\alpha \in \pi\_n(X, x\_0)$ be represented by a map $D^n\_k \to X$ sending the boundary to the constant at $x\_0$?
2. If not, is there a better definition of subdivided cubes for which the answer to (1) becomes "yes"?


It's nice that with the above definition, $D^n\_{k+1}$ can be obtained by gluing together a bunch of copies of $D^n\_k$ in an easy way. But perhaps this is too good to be true.


A trivial observation is that for $n=1$ the above cubical subdivision is basically the same as barycentric subdivision and the answer to (1) comes out as "yes".



Representing simplicial homotopy classes cubically?
I've worked out a proof that a form of cubical $Ex^\infty$ functor is a fibrant replacement functor for cubical sets with connections. From this, it's basically immediate that a homotopy class can be represented as above. I hope to post this to the arxiv soon.


