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Frobenius algebras and ambidextrous adjunctions

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Aaron D. Lauda

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 $\mathcal{D}$ 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 $\mathcal{D}$ 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.

Keywords:
pseudoadjunction, pseudomonad, Frobenius pseudomonoid,
Eilenberg-Moore completion

2000 MSC:
18D05, 57R56, 18D35, 57Q20

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 4, pp 84-122.

http://www.tac.mta.ca/tac/volumes/16/4/16-04.dvi

http://www.tac.mta.ca/tac/volumes/16/4/16-04.ps

http://www.tac.mta.ca/tac/volumes/16/4/16-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/4/16-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/4/16-04.ps

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