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Frobenius Algebras and ambidextrous adjunctions (2006)

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by Aaron D. Lauda
Citations:8 - 0 self
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BibTeX

@MISC{Lauda06frobeniusalgebras,
    author = {Aaron D. Lauda},
    title = {Frobenius Algebras and ambidextrous adjunctions},
    year = {2006}
}

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Abstract

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-categoryDinto which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2-category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2-category and utilizing the free completion under Eilenberg-Moore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2-category M, and replacing the 2-categoryD into which it embeds by a semistrict 3-category. To state this more powerful result, we must first define the notion of a ‘Frobenius pseudomonoid’, which categorifies that of a Frobenius object. We then define the notion of a ‘pseudo ambijunction’, categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2-category arises from a pseudo ambijunction in some semistrict 3-category.

Citations

109 The formal theory of monads - Street - 1972
39 From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories - Müger
27 Limits indexed by category-valued 2-functors - Street - 1976
22 Fibrations and Yoneda’s lemma in a 2-category - Street
21 Ordinal sums and equational doctrines - Lawvere - 1969
16 Enriched Categories, Internal Categories, and Change of Base - Verity - 1992
15 Frobenius algebras and planar open string topological field theories. arXiv:math.QA/0508349 - Lauda - 2005
14 Doctrines whose structure forms a fully faithful adjoint string - Marmolejo - 1997
14 Low-dimensional topology and higher-order categories - Street - 1995
12 Frobenius monads and pseudomonoids - Street
10 S-structures for k-linear categories and the definition of a modular functor - Tillmann - 1998
9 Distributive laws for pseudomonads - Marmolejo - 1999
2 Homologie des anneaux et des modules - Lane - 1956
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