This question is motivated by my attempt at proving that
Proposition: let $X$ be a simplicial set and $x,y \in X_n$. If $x$ and $y$ are homotopic, then $\partial x = \partial y$.
where $\partial z = (d_0z,\dots,d_nz)$.
The references I am using are May's Simplicial Objects in Algebraic Topology and Goerss and Jardine's
Simplicial Homotopy Theory.
In the latter, a homotopy of maps $h : f \simeq g$ is defined as a simplicial map $h : X \times \Delta^1 \to Y$ such that $hi_0 = f$ and $hi_1 =g$ where $i_j$ are the inclusions of $X$ into $X \times \Delta^1$ via
$$X \simeq X \times \Delta^0 \xrightarrow{(1,d^j)} X \times \Delta^1.$$
Two simplices $x,y \in X_n \subset X$ are homotopic if the maps $x,y : \Delta^n \to X$ induced by these are homotopic.
On the other hand May's book says that two simplices $x$ and $y$ are homotopic if $\partial x = \partial y$ and there exists $z \in X_{n+1}$ such that $d_nz =x, d_{n+1}z = y$ and $d_i z = s_{n-1}d_ix = s_{n-1}d_iy$ for each $i$. Similarly, a homotopy between maps $f,g :X \to Y$ is a collection of maps $h_i : X_q \to Y_{q+1}$ for each $0 \leq i \leq q$ that verify a series of combinatorial identities.
How are these two equivalent definitions related? And more concretely, is there a succint proof of the fact that the boundaries of homotopic simplices coincide from the more 'categorical' definition of homotopy?
Best Answer
That's not true, even for Kan complexes. Indeed take a topological space $X$ and its singular simplicial set $Sing(X)$. Then a simplex $x\in Sing(X)$ is a map $x: |\Delta^n| \to X$.
Now suppose you have two simplices $x,y\in Sing(X)$. A homotopy in May's sense between the two is a map $\Delta^n\times \Delta^1\to Sing(X)$ satisfying the appropriate things, and under the $|-|\dashv Sing$ adjunction this translates to a continuous map $|\Delta^n|\times |\Delta^1|\cong |\Delta^n\times \Delta^1|\to X$ satisfying the obvious thing.
But now this is just a topological homotopy between $x,y: |\Delta^n|\to X$, which can of course not respect the boundary of $|\Delta^n|$, in particular we can clearly have $\partial x \neq \partial y$ (take for instance $X=\mathbb R^2$ and a homotopy that crushes the standard $|\Delta^2|\subset \mathbb R^2$ to a point )
I think they defined different notions with different purposes : May defines homotopy rel $\partial \Delta^n$, so as to define homotopy groups later on, whereas Jardine just defines homotopy in a general sense.
In particular you will not be able to prove the equivalence.
What you will be able to prove is something along those lines : if $h:\Delta^n\times \Delta^1\to X$ is a homotopy between $x,y$ whose restriction to $\partial\Delta^n\times \Delta^1$ factors through the projection $\partial \Delta^n\times \Delta^1\to \partial \Delta^n$, then $\partial x = \partial y$ and if $X$ is a Kan complex, then $x,y$ are "May-homotopic".
Moreover, if $x,y$ are "May-homotopic", then they are "Jardine-homotopic" rel $\partial \Delta^n$