I am cross-posting this wuestion from [mathSE](https://math.stackexchange.com/q/1931718/60713). I hope it is of an adequate level for MO.


Let $K$ be a Kan complex, let $f,g:K\to K$ be morphisms of simplicial sets. Consider a simplex $\sigma\in K\_n$, and let $\alpha$ be a vertex of $\sigma$. Assume I have a $1$-simplex $\gamma\in K\_1$ such that $\partial\_0\gamma = f(\alpha)$ and $\partial\_1\gamma = g(\alpha)$. I will describe a way to construct a morphism
$$h\_\sigma\in\hom(\Delta^1\times\Delta^n,K)$$
such that the restrictions of $h\_\sigma$ to $\{0\}\times\Delta^n$ and $\{1\}\times\Delta^n$ correspond to the simplices $f(\sigma)$ and $g(\sigma)$ respectively from this data. I will implicitly use the well known fact that
$$\hom(\Delta^p,K)\cong K\_p$$
(coming from the Yoneda lemma) in what follows.


First, we give an explicit description of $\Delta^1\times\Delta^n$. Consider the poset
$$P\_n = \{[i,j]\mid i\in\{0,1\},0\le k\le n\}$$
with $[i\_1,j\_1]<[i\_2,j\_2]$ if, and only if $i\_1\le i\_2$, $j\_1\le j\_2$, and at least one of the two inequalities is strict. For example, if $n=2$ the poset can be represented by
$$\require{AMScd}
\begin{CD}
[0,1] @>>> [1,1] @>>> [2,1]\\
@AAA @AAA @AAA\\
[0,0] @>>> [1,0] @>>> [2,0]
\end{CD}$$
Then non-degenerate $k$-simplices in $\Delta^1\times\Delta^n$ are represented by strictly increasing $k$-tuples in $P\_n$, and faces are obtained by deleting an element in a tuple. For details, see for example [here](https://www.researchgate.net/publication/292155377_Quillen_equivalence_of_topological_spaces_and_simplicial_sets) (page 45).


Without loss of generality, let $\alpha$ be the $0$-th vertex of $\sigma$. We construct $h\_\sigma$ by describing it on non-degenerate simplices, using the data we have as starting point and relying heavily on the horn-filling property. Therefore, a maximal non-degenerate simplex will be an $(n+1)$-tuple of the form
$$([0,0],[1,0],\ldots,[i,0],[i,1],\ldots,[n,1]).$$
We order these simplices depending on the $i$.


Of course, we define
\begin{align}
h\_\sigma(([0,0],\ldots,[n,0])) = & f(\sigma),\\
h\_\sigma(([0,1],\ldots,[n,1])) = & g(\sigma),\\
h\_\sigma(([0,0],[0,1])) = & \gamma.
\end{align}
Now consider the first of the maximal simplices, given by
$$([0,0],[0,1],\ldots,[n,1]).$$
Notice that we have $h\_\sigma(([0,0],[0,1]))$ and $h\_\sigma(([0,1],[k,1]))$ for all $k$ (given by a $1$-simplex in $g(\sigma)$). But this is exactly the data of a horn of the simplex $([0,0],[0,1],[k,1])$, so we get a filler for all these simplices thanks to the fact that $K$ is a Kan complex. Using this data, we can similarly fill $3$-simplices, then $4$-simplices, and so on, up to filling the whole $(n+1)$-simplex.


Some of the data thus obtained allows us to start a similar process for the second $(n+1)$-simplex, filling it, and so on up to filling the whole $\Delta^1\times\Delta^n$. I can make this more precise if needed.



> 
> **Questions:**
> 
> 
> 1) Has something similar already been done? Well, I'm pretty sure that the answer to this one is yes, but does anybody have a nice reference?
> 
> 
> 2) Can this be used to construct a (simplicial) homotopy between $f$ and $g$ if we are given a $\gamma$ for each $0$-simplex in $K$? (I think we might have some kind of "functoriality" problems here. Namely, when we do our filling process we obtain the $1$-simplices $([i,0],[i,1])$, which might not agree with the corresponding $\gamma$. Is there a way to solve this problem?)
> 
> 
> 



Constructing a homotopy from some starting data
The answer to your question (1) is yes. What you're doing is showing that this inclusion
$$(\{0,1\} \times \Delta^n) \cup (\Delta^1 \times \{\alpha\}) \to \Delta^1 \times \Delta^n,$$
as an iterated pushout of horn-fillers, is an anodyne extension -- a special type of acyclic cofibration. The property of being an acyclic cofibration would also give you the extension property for maps into Kan complexes by Quillen's construction of a model structure on simplicial sets. (It's much easier to check that this map is an acyclic cofibration, since it's an inclusion between simplicial sets whose geometric realizations are both contractible.)


This map is also the so-called pushout-product of the cofibration $\{0,1\} \to \Delta^1$ and the acyclic cofibration $\{\alpha\} \to \Delta^n$, and so a reference which shows that simplicial sets form a so-called "monoidal model category" will also imply that this map is an acyclic cofibration.


(Sorry I don't have a reference for this exact fact for you.)


The answer to your question (2) is no. For example, if $K$ is path-connected then it is always possible to choose a path $\gamma$ between $f(\alpha)$ and $g(\alpha)$ for each 0-simplex $\alpha$ in $K$. A positive answer to (2) would imply that any two maps $f$ and $g$ from $K$ to itself were homotopic. In particular, the identity map would be homotopic to the trivial map, and this only happens if $K$ is contractible. As a concrete example, you could take a group $G$ and consider the identity map and the trivial map from the standard classifying simplicial set $BG$ to itself. Then $BG$ even only has one zero-simplex, and any 1-simplex $\gamma \in G$ can be used as your path.


