Question about faces of a simplex and relation to complexes

Question

I'm reading Hatcher's book on algebraic topology, p103:

Let $[v_0, \dots, v_n]$ be an $n$-simplex. A face of $[v_0, \dots, v_n]$ is the $(n-1)$-simplex obtained by deleting one vertex $v_i$ from the given $n$-simplex.

Hatcher adopts the following convention:

The vertices of any face will always be ordered according to their order in the larger simplex.

There is a special $n$-simplex $\Delta^n:= \{(t_0, \dots, t_n): \sum_i t_i = 1, t_0, \dots, t_n \geq 0 \}$ and a canonical linear homeomorphism

$$\Delta^n \to [v_0, \dots, v_n]: (t_0, \dots, t_n) \mapsto \sum_i t_i v_i$$

A $\Delta$-complex on a space $X$ is a collections of maps $\sigma_\alpha: \Delta^n \to X$ such that

(i) ....

(ii) Each restriction of $\sigma_\alpha$ to a face of $\Delta^n$ is one of the maps $\sigma_\beta: \Delta^{n-1} \to X$. Here we are identifying the face of $\Delta^n$ with $\Delta^{n-1}$ by the canonical linear homeomorphism between them that preserves the ordering of the vertices.

(iii) ...

Question: How does this identification work? Say I have the map $\sigma_\alpha$ and the n-simplex $\Delta^n$, which I can write as

$$\Delta^n := [e_0, \dots, e_n]$$ with $e_0, \dots, e_n$ the canonial basis of $\mathbb{R}^{n+1}$. For example, restrict $\sigma_\alpha$ to the face we get by leaving out $e_2$. Thus we end up with the face $[e_0, e_1, e_3, \dots, e_n]$.

Consider the canonical homeomorphism $\psi: \Delta^{n-1}\to [e_0, e_1, e_3, \dots, e_n]$ sending (by abuse of using the same notation for the basis vectors) $e_0 \mapsto e_0, e_1 \mapsto e_1, e_2 \mapsto e_3, e_3 \mapsto e_4, \dots$

Does this mean that there must be a map $\sigma_\beta: \Delta^{n-1} \to X$ in our collection such that $\sigma_\beta \circ \psi^{-1} =\sigma_\alpha\vert_{[e_0,e_1, e_3, \dots, e_n]}$?

Have I understood how this works correctly?

Answer 1

What he means is simple enough to state without the "I"-word, "identification".

Some notation (I'm not attempting to be standard here, just making up some notation and borrowing your $\psi$ notation): for $i=0,\ldots,n$, the $i^{\text{th}}$ face of $\Delta^n$ is denoted $$F_i \Delta^n $$ and the canonical map from $\Delta^{n-1}$ to $F_i \Delta^n$ is denoted $$\psi^{n-1}_i : \Delta^{n-1} \to F_i \Delta^n $$ Property (ii) simply means:

• For each $\sigma_\alpha : \Delta^n \to X$ and each $i=0,\ldots,n$ there exists $\sigma_\beta : \Delta^{n-1} \to X$ such that $$\sigma_\beta = \sigma_\alpha \circ \psi^{n-1}_i $$