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.
Read more about this topic: Monoidal Category
Famous quotes containing the words free, strict and/or category:
“Let those who desire a secure homeland conquer it. Let those who do not conquer it live under the whip and in exile, watched over like wild animals, cast from one country to another, concealing the death of their souls with a beggar’s smile from the scorn of free men.”
—José Martí (1853–1895)
“The admission of Oriental immigrants who cannot be amalgamated with our people has been made the subject either of prohibitory clauses in our treaties and statutes or of strict administrative regulations secured by diplomatic negotiations. I sincerely hope that we may continue to minimize the evils likely to arise from such immigration without unnecessary friction and by mutual concessions between self-respecting governments.”
—William Howard Taft (1857–1930)
“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)