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:
“But we still remember ... above all, the cool, free aspect of the wild apple trees, generously proffering their fruit to us, though still green and crude,—the hard, round, glossy fruit, which, if not ripe, still was not poison, but New English too, brought hither, its ancestors, by ours once. These gentler trees imparted a half-civilized and twilight aspect to the otherwise barbarian land.”
—Henry David Thoreau (1817–1862)
“Governments which have a regard to the common interest are constituted in accordance with strict principles of justice, and are therefore true forms; but those which regard only the interest of the rulers are all defective and perverted forms, for they are despotic, whereas a state is a community of freemen.”
—Aristotle (384–322 B.C.)
“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)