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].
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.
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]
Last revised on August 23, 2023 at 08:31:27. See the history of this page for a list of all contributions to it.