Let $K$ be a pure simplicial complex of dimension $d$. I would like to ask, if there is a way to describe a simplicial complex by means of certain subcomplexes rather than by simplices. Suppose I understand that $K$ is constructed from two subcomplexes $A$ and $B$ of dimension $d$ that share a common subcomplex $C$ in its boundary, say of dimension $d'<d$.
For example let $A$ be the cone over a (stellar) subdivided edge $C$, that is $A=C\ast a$, and $B$ be another cone over $C$, that is, $B=C\ast b$, so that $A$ and $B$ coincide in $C$.
I feel that this information should suffice to fully describe $K$ up to simplicial isomorphism.
My question is, how can I formalize this information/construction within the context of the usual definition of a simplicial complex? To give the full set of simplices, which one has to do be definition, seems superfluous here.
You may be looking for the pushouts of simplicial complexes. Here is a recipe to compute them in the special case of working with subcomplexes.
Recall that a simplicial complex $X$ is the data of a set $V(X)$ of vertices together with a downward-closed family $S(X)$ of subsets of $V(X)$. Given a span $$A \hookleftarrow C \hookrightarrow B$$ of inclusions of simplicial complexes, the pushout $A \cup_C B$ has vertices $$V(A \cup_C B) = V(A) \cup_{V(C)} V(B)$$ and simplices $$S(A \cup_C B) = S(A) \cup S(B).$$
However, while the pushout still exists if $C$ is not a subcomplex of $A$ and $B$, its description is more complicated. Moreover, it is not true that geometric realization preserves these pushouts in general, so pushouts in simplicial complexes and the corresponding pushouts in topological spaces could look different.