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
A functor between monoidal categories which is both a lax monoidal functor and an oplax monoidal functor is called Frobenius if the structure morphisms for the lax monoidal structure
and for the oplax monoidal structure
satisfy the axioms of a Frobenius algebra in that (in string diagram notation) for all objects in we have
and
The Moore complex functor
from abelian simplicial groups to connective chain complexes is Frobenius, as is the normalzed chains complex functor
For more on this see monoidal Dold-Kan correspondence.
Frobenius monoidal functors are a special case of Frobenius linear functors between linearly distributive categories.
Frobenius monoidal functor
This concept is related to that of
albeit in a nontrivial way. See Theorem 3.18 of (Flake et al 2024) for more on this.
Marcelo Aguiar, Swapneel Mahajan, (3.26), (3.27) on p. 81 of: Monoidal Functors, Species and Hopf Algebras, CRM Monograph Series 29, Amer. Math. Soc. (2010) [ISBN:978-0-8218-4776-3, pdf]
M. B. McCurdy, R. Street, What separable Frobenius monoidal functors preserve, Cahiers TGDC 51 (2010) 29-50 [arxiv/0904.3449] .
Johannes Flake, Robert Laugwitz, Sebastian Posur. Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers (2024). (arXiv:2410.08702).
Relation to Drinfeld centers:
Last revised on November 15, 2024 at 18:00:56. See the history of this page for a list of all contributions to it.