# nLab enriched monoidal category

Contents

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

The notion of enriched monoidal categories is a compatible combination of the notions of enriched categories and monoidal categories. The main point is that the tensor product-functor on the underlying monoidal category is properly an enriched functor with respect to the underlying enriched category.

Special cases include tensor categories, which are (Vect,$\otimes$)-enriched monoidal categories.

## Definition

### The monoidal 2-category of enriched categories

For $V$ a symmetric monoidal cosmos, the 2-category VCat of $V$-enriched categories becomes a monoidal 2-category by declaring the tensor product $\mathbf{C} \otimes \mathbf{D}$ of a pair $\mathbf{C}$, $\mathbf{D}$ of $V$-enriched categories to have

• as set of objects the cartesian product of the given sets of objects:

$Obj(\mathbf{C} \otimes \mathbf{D}) \;\coloneqq\; Obj(\mathbf{C}) \times Obj(\mathbf{D})$
• as hom-objects the tensor product in $V$ between the given hom-objects:

$(\mathbf{C} \otimes \mathbf{D})\big((c,d), (c',d')\big) \;\coloneqq\; \mathbf{C}(c,c') \otimes \mathbf{D}(d,d') \,.$
• composition obtained by the given composition operations, after using the braiding in $V$ to align factors:

$\array{ (\mathbf{C} \otimes \mathbf{D}) \big((c',d'), (c'', d'')\big) \otimes (\mathbf{C} \otimes \mathbf{D}) \big((c,d), (c',d')\big) &\overset {\circ_{\mathbf{C} \otimes \mathbf{D}}}{ \longrightarrow } & (\mathbf{C} \otimes \mathbf{D}) \big((c,d), (c'',d'')\big) \\ \mathllap{{}^\simeq}\Big\downarrow \phantom{---------} && \Big\uparrow\mathrlap{{}^{\simeq}} \\ \mathbf{C}(c',c'') \otimes \mathbf{D}(d',d'') \otimes \mathbf{C}(c,c') \otimes \mathbf{D}(d,d') \underoverset {braid} {\sim} {\to} \mathbf{C}(c',c'') \otimes \mathbf{C}(c,c') \otimes \mathbf{D}(d',d'') \otimes \mathbf{D}(d,d') &\overset{ \circ_{\mathbf{C}} \otimes \circ_{\mathbf{D}} }{\longrightarrow}& \mathbf{C}(c,c'') \otimes \mathbf{D}(d,d'') }$

### Enriched monoidal categories

A $V$-enriched monoidal category $\mathbf{C}$ is a pseudomonoid internal to the above monoidal 2-category $V Cat$.

This means mainly that

• $\mathbf{C}$ is a $V$-enriched category

• whose underlying category $C$ is equipped with monoidal category-structure, hence with a tensor product-functor

$\otimes \,\colon\, C \times C \longrightarrow C$
• which is compatibly lifted for each pair of pairs of objects of $C$ to a morphisms on hom-objects in $V$:

$\otimes_{\mathbf{C}}\big( (c,d),\, (c',c') \big) \,\colon\, \mathbf{C}(c,c') \otimes \mathbf{C}(d,d') \longrightarrow \mathbf{C}(c \otimes_C d ,\, c' \otimes_C d')$

in a compatible way.

(…)

## References

Last revised on May 31, 2023 at 12:21:12. See the history of this page for a list of all contributions to it.