Monoidal functor

Let $𝖢, 𝖣$ be monoidal categories.¹ A functor $𝑇 : 𝖢 \to 𝖣$ is called monoidal iff it is eqquipped with an isomorphism $𝜈 : 𝟙 \to 𝑇 𝟙$ in $𝖣$ and a natural isomorphism with components $𝛾 𝑥, 𝑦 : 𝑇 𝑥 \otimes 𝑇 𝑦 \to 𝑇 (𝑥 \otimes 𝑦)$ in $𝖣 𝖢 \times 𝖢$ , compatible with associativity

and unitality

Iff $𝜖$ and $𝜇$ are identities, then $𝑇$ is called strict monoidal.² If $𝖢$ and $𝖣$ are braided, then a monoidal functor $𝐹 : 𝖢 \to 𝖣$ is said to be braided iff

commutes for all objects $𝑥, 𝑦 \in 𝖢$ .

Examples

Free module

Monoidal functor

Examples

See also

Footnotes