With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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,)-enriched monoidal categories.
For a symmetric monoidal cosmos, the 2-category VCat of -enriched categories becomes a monoidal 2-category by declaring the tensor product of a pair , of -enriched categories to have
as set of objects the cartesian product of the given sets of objects:
as hom-objects the tensor product in between the given hom-objects:
composition obtained by the given composition operations, after using the braiding in to align factors:
A -enriched monoidal category is a pseudomonoid internal to the above monoidal 2-category .
This means mainly that
is a -enriched category
whose underlying category is equipped with monoidal category-structure, hence with a tensor product-functor
which is compatibly lifted for each pair of pairs of objects of to a morphisms on hom-objects in :
in a compatible way.
(…)
cartesian closed enriched category, locally cartesian closed enriched category
enriched monoidal model category, simplicial monoidal model category
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]
Last revised on June 1, 2024 at 14:38:40. See the history of this page for a list of all contributions to it.