The Catalan simplicial set
arXiv:1309.6120 · doi:10.1017/S0305004114000498
Abstract
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.
15 pages. Replaces and expands upon parts of arXiv:1307.0265; remaining parts of arXiv:1307.0265 will be incorporated into a sequel. Version 2: minor revision; to appear in Math. Proc. Camb. Phil. Soc