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:
“The majority is never right. Never, I tell you! That’s one of these lies in society that no free and intelligent man can help rebelling against. Who are the people that make up the biggest proportion of the population—the intelligent ones or the fools? I think we can agree it’s the fools, no matter where you go in this world, it’s the fools that form the overwhelming majority.”
—Henrik Ibsen (1828–1906)
“The right honourable gentleman caught the Whigs bathing, and walked away with their clothes. He has left them in the full enjoyment of their liberal positions, and he is himself a strict conservative of their garments.”
—Benjamin Disraeli (1804–1881)
“I see no reason for calling my work poetry except that there is no other category in which to put it.”
—Marianne Moore (1887–1972)