Let $A, X$ be simplicial sets, and suppose there's an inclusion $A \longrightarrow X$. Geometrically realizing the inclusion map, we get a pair of spaces $(\mathcal{A}, \mathcal{X})$. I want to find a simplicial set whose geometric realization has the homotopy type of $\mathcal{X} \setminus \mathcal{A}$.

Here are my thoughts so far:

- If we try to remove from $X$ all simplices that are also in $A$, we run into immediate trouble: other simplices in $X$ probably have simplices of $A$ as their faces. Somehow we need to remove the simplices of $A$ and then adjoin new boundary simplices ``as freely as possible.''
- Ideally such a construction would be functorial. There's no way that the domain can be the category of pairs, however, since there won't be induced maps after subtracting. Instead, it seems like the source category should be more like a twisted arrow category (although this doesn't seem to work either).

The case I'd really like to get working is where $X$ is a $G$-simplicial set with finitely-many non-degenerate simplices in each degree and $A$ is the points which have nontrivial stabilizer, but I'm interested in the general case as well.