nLab
Frobenius monoidal functor
Contents
Context
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
Definition
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
Examples
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.
Generalizations
Frobenius monoidal functors are a special case of Frobenius linear functors between linearly distributive categories.
References
Equation (3.26), (3.27) in p. 81 of
- M. B. McCurdy, R. Street, What separable Frobenius monoidal functors preserve, arxiv/0904.3449 and Cahiers TGDC, 51 (2010)p. 29 - 50.
Last revised on October 20, 2017 at 17:16:57.
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