A Picard 2-category is a 2-category equipped with a group-like structure that has commutativity-like properties. An additive 2-functor between two Picard 2-categories is a 2-functor that respects the group-like structures.

Let A be a Picard 2-stack. We prove that any set map into A extends to an addtive 2-functor from the free abelian group generated by the set into A. This fact is the analog of the universal property of free abelian groups.

In this talk, we give the highlights of the combinatorial proof of this fact such as lattice paths and permuto-associahedron.