Complex of $n$-balls in analogy of simplicial complexes

Question

I would like to know if there is a version of complexes (as in simplicial complexes/sets) with the role of simplices played by $n$-balls.

I thought of the following:

1. the analogue of a (-1)-simplex(?) should be a circle ()
2. the analogue of a 0-simplex should be a disk (0) bounded by the circle ()
3. the analogue of a 1-simplex should be a 3-ball (0,1) bounded by two disks (0) and (1) glued together by the circle ()
4. the analogue of a 2-simplex should be a 4-ball(??) (0,1,2) bounded by three 3-balls (0,1), (0,2) and (1,2) glued together by the respective common disks (0), (1) and (2) which in turn are glued together by the common disk ()
5. ...

I do realize that this seems like an unnecessarily complicated geometric model of the simplex category. Basically the only difference is that this geometric model starts one level lower than simplices without any degeneracies, so that the empty set has an actual geometric meaning. If you have a better solution for this problem, you're welcome to share it ;)

My questions are: Does this model makes any sense at all (well-defined, etc)? If so, is there maybe already a rigorous definition of this in the sense of a $n$-ball category in analogy to the simplex category?

Answer 1

What you are describing is known as a globular set.

The article referenced in the link contains more information, including the pertinent literature.