nLab tensor product of enriched categories




For (𝒱,)(\mathcal{V}, \otimes) a symmetric monoidal category regarded as a cosmos for enrichment, there is a tensor product of 𝒱 \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 𝒱\mathcal{V} to align composable tensor factors.

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

If 𝒱\mathcal{V} is furthermore closed and complete so that 𝒱 \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 𝒱=\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:

See also

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

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

Last revised on January 31, 2024 at 09:37:47. See the history of this page for a list of all contributions to it.