The short answer is: the maps $\partial$ are defined on the free abelian groups generated by $n$-simplices, hence it makes sense to speak about sums of simplices in these groups.
I'll try to give a down to earth answer, for which we will need to detour first. Consider the set $\mathbb{Z}[X]$ of integer polynomials. An element there is an expression
$$
a_0 + a_1 X + \ldots + a_nX^n
$$
where each $a_i$ is an integer, and $X$ is an indeterminate. Even though we might be quite familiar with thinking of polynomials in this way, the terms 'expression' and 'indeterminate' are informal. A rigorous definition of $\mathbb{Z}[X]$ could be the set of sequences of integers that are eventually zero,
$$
\mathbb{Z}[X] = \{(a_n)_{n \geq 0} \in \mathbb{Z}^{\mathbb{N}_0} : \text{there exists $k \in \mathbb{N}_0$ such that $a_n = 0$ for all $k \geq n$}\}. \tag{1}
$$
Given integers $a_1,\ldots,a_n\in \mathbb{Z}$, we then define
$$
a_0 + a_1 X + \ldots + a_nX^n := (a_0,a_1,\ldots,a_n,0,0,\ldots) \tag{2}.
$$
All the operations on polynomials can be defined in terms of sequences as in $(1)$ and are compatible with the usual notation $(2)$. This is a way to formally capture the idea of having some unknown element $X$ that can be 'scaled' by integers, multiplied by itself, and such that we can make sense of the sums of expressions like this.
In the same spirit, suppose you have a set $X$ and you want to have another set $M$ such that:
- $X$ is, in some sense, contained in $M$,
- we can make sense of multiplying $x \in X$ by an integer $k \in \mathbb{Z}$,
- we can give meaning to an expression like $17x+28y+3z$ for some $x,y,z \in X$.
A way to do this is to define the free abelian group $\mathbb{Z}^{(X)}$ with basis $X$. This concept is closely related to bases of vector spaces. The definition (technically, one definition, there are 'many' equivalent ones) is as follows: as a set $\mathbb{Z}^{(X)}$ consists of functions $f \colon X \to \mathbb{Z}$ such that $f(x) \neq 0$ for finitely many $x \in X$. In other words, the elements of $\mathbb{Z}^{(X)}$ are finitely supported functions $\mathbb{Z} \to X$. We can make sense of sum and multiplication of elements here in the same way we do to define a vector space structure on $\mathbb{R}^X$. Namely:
- the sum of $f,g \in \mathbb{Z}^{(X)}$ is the function $(f+g)(x) := f(x)+g(x)$.
- if $k$ is an integer and $f \in \mathbb{Z}^{(X)}$, we define $(k\cdot f)(x) := kf(x)$.
You can check that these operations define once again elements of $\mathbb{Z}^{(X)}$.
Now, let's go back to the analogy with polynomials; in particular, to the relation between definition $(1)$ and notation $(2)$. Even though formally the free abelian group is defined as finitely supported functions, we want to think of it as "$\mathbb{Z}$-linear combinations of $X$". To do this, we write
$$
x := \chi_{\{x\}}, \quad \chi_{\{x\}}(y) = \begin{cases}1 &\text{if $x=y$}\\
0 &\text{otherwise}\end{cases}
$$
You can check that, following this notation, every element of $\mathbb{Z}^{(X)}$ can be written as a finite sum
$$
a_1 x_1 + \ldots +a_n x_n
$$
for some $x_i \in X$ and integers $a_i$. Moreover, two such expressions are equal if the elements of $X$ appearing are the same, and the coefficients accompanying each element coincide.
In this case, we consider $\mathbb{Z}^{(C_n)}$ with $C_n$ the set of $n$-simplices. In $\mathbb{Z}^{(C_n)}$ it makes perfect sense to speak of $3 \sigma + 22 \tau$ for some pair of simplices $\sigma,\tau$ or any expression of this sort; likewise the maps $\partial_k$ are well defined.
I hope this gives a little bit more context to the first sentence of this answer.
Best Answer
Thanks for your curiosity and nice pictures. I don't have much time but I'm working on similar stuff right now so I'll drop something.
I think your notation is a bit off. Your $V$ is not a simplex. Note that $\Delta_n$ is intented to be a fixed object that only depends on $n$.
It has to do as well with the size (I assume by this you mean dimension) of the simplices you add, the number and their configuration. A simplicial complex with $n$ vertices can always be embedded in $\mathbb{R}^n$. Menger proved in 1928 that every $d$-dimensional simplicial complex can be embedded in $\mathbb{R}^{2d+1}$.
Indeed, embedding twodimensional complexes in $\mathbb{R}^3$ is related to planar graphs, because in general we have the following. Let $\Sigma$ be a simplicial complex and $\mathbf{v}$ a vertex of $\Sigma$. Consider the subcomplex $\Sigma'$ of $\Sigma$ consisting of all those cells of $\Sigma$ that donot contain $\mathbf{v}$. If $\Sigma'$ can be embedded in $\mathbb{R}^d$, then $\Sigma$ can be embedded in $\mathbb{R}^{d+1}$.
Substantial work has been done on understanding which complexes can be embedded in which dimensions of Euclidean space. But it's not easy. You might want to have a look at the following freely available papers: