nLab tensor product of enriched categories

Idea

For $(\mathcal{V}, \otimes)$ a symmetric monoidal category regarded as a cosmos for enrichment, there is a tensor product of $\mathcal{V}$-enriched categories which on classes of objects is the Cartesian product and on hom-objects is given by $\otimes$, with composition defined factor-wise after using the braiding in $\mathcal{V}$ to align composable tensor factors.

This operation makes $\mathcal{V}Cat$ itself into a (very large) symmetric monoidal category [Kelly (1982), §1.4].

If $\mathcal{V}$ is furthermore closed and complete so that $\mathcal{V}$-enriched functor categories exist, then these constitute the internal hom right adjoint to the operation of forming the enriched product category with a given enriched category [Kelly (1982), §2.3].

Examples

Example

For $\mathcal{V} =$ Set equipped with its cartesian product $\otimes \,\coloneqq\, \times$, the construction of enriched product categories reduces to the ordinary notion of product categories.

Related concepts

• bifunctor

References

Textbook accounts:

• Samuel Eilenberg, G. Max Kelly, §III.3 of: Closed Categories, in: Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

• Max Kelly, §2.3 of: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]

See also

• Enrico Ghiorzi, §3.6 Internal enriched categories, Applied Categorical Structures 30 (2022) 947–968 [arXiv:2006.07997, doi:10.1007/s10485-022-09678-w]

• Haskell: Data-Category-Enriched.html

The tensor product of enriched categories is called the commuting tensor product of enriched categories in:

• Richard Garner, Ignacio López Franco?, Commutativity [arXiv:1507.08710]

where the authors show how to construct the tensor product even when $V$ is not braided monoidal, but merely duoidal.