From: Aaron Lauda <lauda@math.columbia.edu>
To: categories@mta.ca
Subject: pivotal adjoints?
Date: Thu, 19 Jul 2007 14:05:46 -0400 [thread overview]
Message-ID: <E1IByam-0002vb-FY@mailserv.mta.ca> (raw)
Dear category theorists,
Suppose we have chosen left and right adjoints for F:A->B and G:A->B
F-| F* -| F and G-| G*-|G
i_F: 1_B => FF* i_G: 1_B => GG*
e_F: F*F => 1_A e_G: G*G => 1_A
j_F: 1_A => F*F j_G: 1_A => G*G
k_F: FF* => 1_B k_G: GG* => 1_B
Then given any 2-morphism a:F=>G there are two obvious duals (mates
under adjunction) for the 2-morphsism a
a+ :G*=>F* := (e_G F*).(G*aF*).(G* i_F)
+a :G*=>F* := (F* k_G).(F*aG*).(j_F G*)
or for those who like pictures:
+a a+
__ __
/ \ | | / \
| | | | | |
| a | | a |
| | | | | |
| \__/ \__/ |
| |
In general a+ is not equal to +a because if is was we could always twist one
of the units and counits so that it does not hold. Has the condition
that a+ = +a been investigated in the literature anywhere? In
particular, if a 2-category is such that all 1-morphisms F have a
simultaneous left and right adjoint then has anyone studied the
context where the adjoints are such that a+ = +a is always
satisfied? Perhaps, this notion has been studied in the language of
duals for 1-morphisms?
The above condition appears to be related to the notion of pivotal
category when we look at Hom(A,A) for any object A.
Thanks,
Aaron Lauda
next reply other threads:[~2007-07-19 18:05 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2007-07-19 18:05 Aaron Lauda [this message]
2007-07-25 11:45 John Baez
2007-07-25 13:27 Aaron Lauda
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