nLab enriched monoidal category

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.

Examples

(…)

Related concepts

• cartesian closed enriched category, locally cartesian closed enriched category

• enriched category, monoidal category

• enriched monoidal model category, simplicial monoidal model category

References

• Jacob Lurie, Section 1.6 in: Derived Algebraic Geometry II: Noncommutative Algebra [arXiv:math/0702299]

  which is Def. 4.1.7.7 in Higher Algebra [pdf]

  (focus on sSet-enrichment for simplicial monoidal model categories)

• Michael Batanin, Martin Markl, Section 2 of: Centers and homotopy centers in enriched monoidal categories, Advances in Mathematics 230 4–6 (2012) 1811-1858 [doi:10.1016/j.aim.2012.04.011, arXiv:1109.4084]

• Michael Ching, Def. 1.10 in: Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005) 833-934 [arXiv:math/0501429, doi:10.2140/gt.2005.9.833]

• Rune Haugseng, answer to Definitions of enriched monoidal category, MO:a/315075 (2018)

• Scott Morrison, David Penneys, Monoidal Categories Enriched in Braided Monoidal Categories, International Mathematics Research Notices 2019 11 June 2019 3527–3579 [doi:10.1093/imrn/rnx217, arXiv:1701.00567]

• Liang Kong, Wei Yuan, Zhi-Hao Zhang, Hao Zheng, Enriched monoidal categories I: centers [arXiv:2104.03121]