1$ be an integer, and write $[v_n]:=\{v_0, v_1, \ldots, v_n\}$. We use $K$ to denote a simplicial complex and $\alpha, \sigma$ etc. to denote a simplex of $K$. If $K$ is a simplicial complex on $n+1$ vertices, the set $V(K):=[v_n]$ is the **vertex set** of $K$ or the set of $0$-simplices of $K$. We use $\sigma^{(i)}$ to denote a simplex of dimension $i$, and we write $\tau < \sigma^{(i)}$ to denote any face of $\sigma$ of dimension strictly less than $i$. The number $\dim(\sigma)-\dim(\tau)$ is called the **codimension of $\tau$ with respect to $\sigma$**.

A **simplicial map** $f\colon K \to L$ is a function induced by a map on the vertex sets $f_V\colon V(K)\to V(L)$ with the property that if $\sigma=v_{i_0}v_{i_1}\ldots v_{i_m}$ is a simplex in $K$, then $f(\sigma):=f_V(v_{i_0})f_V(v_{i_1})\ldots f_V(v_{i_m})$ is a simplex of $L$.

If $v_0, v_1, \ldots, v_n$ are all the vertices of a simplex $\sigma$, we will often use the notation $\sigma:=\prod\limits_{i=0}^n v_i$.

\[lem: comp\] If $f\colon A\to B$ and $g\colon B\to C$ are simplicial maps, then $(g\circ f)_V = g_V \circ f_V$.

Consider any $\sigma \in V(A)$. Then, $(g\circ f)_V(\sigma) = (g\circ f)(\sigma)$. Since $f,g$ are simplicial maps, we also know that $\dim f(\sigma) = \dim 