So for graphs we have adjacency matrices to represent the relations between vertices. Could such a generalised object exist for simplicial complexes?
I guess it would need to be some sort of tensor, where restricting to the "first" dimension would give the adjacency matrix for the one skeleton but then the higher dimensional connections are also encoded.
I've been trying to think about what such an object would be, perhaps an $n \times\dots\times n$ array but I don't see how this could also preserve lower dimensional relations too.
Indeed I think if $i$ goes from 1 to the dimension of the complex, you need a set of $n_i \times \cdots \times n_i$ - arrays where the $i$-th array has dimension $i+1$. $n_i$ is the number of $(i-1)$-simplices contained in your complex. If there is a $i$ simplex connecting $i+1$ simplices of lower dimension, define corresponding entry in the $i$-th matrix to be 1 and otherwise 0.
An simple example of a 1-complex can be a triangle with three vertices $\{A,B,C\}$, three edges $\{a,b,c\}$ and one surface. The two matrices would be: $$ X = \begin{matrix} & A & B & C \cr A & 0 & 1 & 1 \cr B & 1 & 0 & 1 \cr C & 1 & 1 & 0 \cr \end{matrix} $$
$$ Y[a]= \begin{matrix} & a & b & c \cr a & 0 & 0 & 0 \cr b & 0 & 0 & 1 \cr c & 0 & 1 & 0 \cr \end{matrix} $$ $$ Y[b]= \begin{matrix} & a & b & c \cr a & 0 & 0 & 1 \cr b & 0 & 0 & 0 \cr c & 1 & 0 & 0 \cr \end{matrix} $$ $$ Y[c]= \begin{matrix} & a & b & c \cr a & 0 & 1 & 0 \cr b & 1 & 0 & 0 \cr c & 0 & 0 & 0 \cr \end{matrix} $$
As you note this graph is complete and nevertheless has many zeros in its second matrix. I don't know if this procedure has been studied at any time and think the approach with incidence matrices is more straight forward.