Monoidal Category - Free Strict Monoidal Category

Free Strict Monoidal Category

For every category C, the free strict monoidal category Σ(C) can be constructed as follows:

  • its objects are lists (finite sequences) A1, ..., An of objects of C;
  • there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m = n, and then the arrows are lists (finite sequences) of arrows f1: A1 → B1, ..., fn: An → Bn of C;
  • the tensor product of two objects A1, ..., An and B1, ..., Bm is the concatenation A1, ..., An, B1, ..., Bm of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists.

This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat.

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