Monoidal category

Monoidal functor

Let be monoidal categories.¹ A functor is called monoidal iff it is eqquipped with an isomorphism in and a natural isomorphism with components in , compatible with associativity

and unitality

Iff and are identities, then is called strict monoidal.² If and are braided, then a monoidal functor is said to be braided iff

commutes for all objects .

Examples

• Free module

See also

• Monoidal natural transformation

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Footnotes

1. As usual we overload and to denote the tensor products and units of both categories. ↩

2. 1966. Closed categories, §II.1, p. 473 ↩