Frobenius algebras and ambidextrous adjunctions

By Aaron D. Lauda. Theory and Applications of Categories, 2005

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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-category D into 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-category D 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/43128dd43f17e4cc7c5fcec6ea433a19af23ff8f

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Citations in the corpus

Categorified Quantum sl(2) and Equivariant Cohomology of Iterated Flag Varieties (2008, result): “A similar result for U(sl2) categorifications, in the context of non-equivariant cohomology, appears in [6, 7.7.2] extending results of Khovanov; the quantum case was considered in [7, Theorem 6.2] and [ 19, Theorem 9.15 ].”
Categorifications from planar diagrammatics (2010, background): “Yet another terminology for a biadjoint pair is ambidextrous adjunction [31].”
A 2-Categories Companion (2007, methodology): “ories were given in [45, 44], or in the equivalent language of pseudomonoids in [11]; for the universal pseudomonad, and the Gray-limit approach to pseudoalgebras, see [32]. For further work see also [8, 56, 40]. 68 STEPHEN LACK 10. Nerves In this section we use ∆ for the “topologists’ delta”, the c”