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 concept of localization of a category for the case of monoidal categories with localization functors being monoidal functors.
On monoidal Bousfield localization of monoidal model categories:
Clark Barwick, Prop. 4.47 in: On left and right model categories and left and right Bousfield localizations, Homology Homotopy Appl. 12 2 (2010) 245-320 [doi:10.4310/hha.2010.v12.n2.a9, euclid:1296223884, subsuming: arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]
Sergey Gorchinskiy, Vladimir Guletskii, Lemma 28 in: Symmetric powers in abstract homotopy categories, Adv. Math., 292 (2016) 707-754 [arXiv:0907.0730, doi:10.1016/j.aim.2016.01.011]
David White, Monoidal Bousfield Localizations and Algebras over Operads, PhD thesis (2014), published in Equivariant Topology and Derived Algebra: A conference in honor of John Greenlees’ 60th birthday, Cambridge University Press (2021) 179-239 [doi:10.1017/9781108942874.007, digitalcollections:ir-2263, arXiv:1404.5197]
Tyler Lawson, Prop. 12.18 in: An introduction to Bousfield localization, in: Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press (2022) [arXiv:2002.03888]
Luca Pol, Jordan Williamson, The Left Localization Principle, completions, and cofree -spectra, J. Pure Appl. Algebra 224 11 (2020) 106408 [arXiv:1910.01410, doi:10.1016/j.jpaa.2020.106408]
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