Frobenius monads and pseudomonoids Introduction - ResearchGate

=
h
d
h
d
h
d
m
m
m
d
e
e
=
=
h
m
d
h
h
d
m
d
=
m
e
d
=
. QED
Remark (a) It follows from the first sentence of Lemma 1.3 that r is uniquely
determined by the monad
T = ( T, h, m) and e . This is because the counit s is
determined by m and e , and the counit of any adjunction uniquely determines the unit.
(b) It is implicit (using AM1) in the second sentence of Lemma 1.3 that d
coassociative.
is
Proposition 1.4 For a Frobenius monad T, the left adjoint
functor
T T
U
: X æÆ æ X is also a right adjoint to
F
U T with counit e.
T T
Proof By AM2 we know that there is an isomorphism of categories K :
that
V G K =
right adjoint
U T . The left adjoint
C G to
V G is given by
F T to
U T is given by
G X
C X = TX ææÆT X
10
d 2
( )
: X æÆ æ X to the forgetful
m
( )
T 2 X
F X = T X ææÆTX T G X @ X such
and the
. Since Tm . rT = d,
we see