Removing a simplicial subset from a simplicial set 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.
 A: Here are some perhaps silly ideas. 
(i) Think of simplicial sets as presheaves on $\Delta$, now look up when a subpresheaf can be subtracted. (E.g. is there a universal property involved?) What os the subobject classified in the presheaf topos here? I am sure that these things are known, but possibly do not have a simple answer. 
(ii) One related point from Peter May's answer is that $X\setminus A$ should be the maximal subsimplicial set of $X$ with trivial intersection with $A$. The  condition on the realisations is then a possible red herring.  
(iii) A final point is that if you are handling a simplicial complex (say a PL manifold, with a given triangulation) there would be questions of subdivision, and if I remember rightly some sort of regular neighbourhoods, and I think that this is geometrically more significant that merely looking at the complement of $|A|$ in $|X|$.
A: The 0-simplices of Sd(X) are the simplices of X.  Then X\A gives a subset of Sd_0(X), how about the subspace of Sd_.(X) spanned by those?
