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: A1B1, ..., fn: AnBn 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.

Read more about this topic:  Monoidal Category

Famous quotes containing the words free, strict and/or category:

    American future lies in the East. The great free markets of the Pacific Rim are the American destiny.
    Donald Freed, U.S. screenwriter, and Arnold M. Stone. Robert Altman. Richard Nixon (Philip Baker Hall)

    To safeguard one’s health at the cost of too strict a diet is a tiresome illness indeed.
    François, Duc De La Rochefoucauld (1613–1680)

    The truth is, no matter how trying they become, babies two and under don’t have the ability to make moral choices, so they can’t be “bad.” That category only exists in the adult mind.
    Anne Cassidy (20th century)