Frobenius monads and pseudomonoids Introduction - ResearchGate

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adjoint, is Frobenius with l = * j . Proof In an autonomous monoidal bicategory, the composite a biexact pairing if and only if the corresponding morphism is an equivalence. So, with an equivalence. QED o Ÿ nƒA o A ƒ s o s : A æææÆA ƒA ƒA æææÆA l = * j , we have ( ) ( ƒ ) ƒ ( ) @ sŸ o * o * = A ƒj o A p o n A d , p s :A ƒA æ æÆA æ æÆI l A pseudocomonoidal structure on A in Proposition 3.1 is provided by j * and p * ; compare Theorem 1.6 (c). We also note that there are isomorphisms * p @ d ƒA o A o p o( n A) A p o r A p A o A r where ( ) ƒ Ê j p* ˆ r= ÁI æÆ æ A ææÆA ƒA˜. Ë ¯ Example Quasi-Hopf algebras ( ) ƒ @ ƒ ( ) ( ƒ ) @ ( ƒ ) ( ƒ ) A quasibialgebra (over a field k) is a k-algebra H equipped with algebra morphisms D :H H H æÆ æ ƒ and E: H æÆ æ k , and with an invertible element fŒH ƒH ƒH, such that ( E ƒ 1H)( D( a) ) = a = ( 1H ƒE)( D( a) ) and f( D ƒ 1H)( D( a) ) = ( 1H ƒ D)( D( a) ) f for all a Œ H; furthermore, f satisfies the pentagon condition ( 1ƒ1ƒ D) ( f) ◊ ( D ƒ1ƒ1) ( f) = ( 1ƒ f) ◊ ( 1ƒ D ƒ1) ( f) ◊( f ƒ 1) . (A quasibialgebra reduces to an ordinary bialgebra when f is the identity element 1 1 1 ƒ ƒ .) We can make H ƒ H actions into a left H ƒ H-, right H-bimodule by means of the ( a ƒ b)◊( x ƒ y)◊ c = 1 2 axc ƒbyc where Â i i Â i i i i 1 2 ; for the time being, let us call this bimodule M. Given an algebra D( c) = c ƒc anti-morphism S: H æÆ æ H, there is another left H ƒ H-, right H-bimodule structure defined on H ƒ H by the actions ( a ƒ b)◊( x ƒ y)◊ c = 1 2 Â axS( bj) ƒbj yc where D( b) = b ƒb j Â j j j 1 2 ; for the time being, let us call this bimodule N. A quasi-Hopf 18 is
algebra is a quasibialgebra H together with an algebra anti-morphism S: H æÆ æ H (called the antipode) and a bimodule isomorphism p :M@ N. This is equivalent to the original definition of [Dd] that, instead of the isomorphism p , involved two elements a H satisfying the equations and b of Ec () a = ÂSc ( i) aci 1 2 , Ec () b = Âci bSc ( i ) 1 2 1 2 3 , Â fi bS( fi ) afi = 1 -1 -2 -3 and Â S( fi ) a fi bS( fi ) = 1 i i i (see Section 2.4 of [Maj]). We shall say a bit more about the equivalence of the definitions soon. Take V = Vect k and recall that a k-algebra A is a one-object V-category SA. It is obvious that an algebra morphism f: A æÆ æ B is the same as a V-functor f: SA æÆ æ SB. So what is a V-natural transformation s :ffi g in terms of the algebra morphisms f and g from A to B ? It is nothing other than an element a of B such that afa ( ) = ga ( ) a for all elements a of A. Therefore we see that f belongs in the square in SH D SH ƒ SH fl f D Dƒ1 SH ƒ SH SH ƒ SH ƒ SH 1 ƒD V - Cat. From this it follows easily that: a quasibialgebra structure on an algebra H is precisely a normalized pseudomonoid structure on SH in V - Cat coop ( ) . Now we move from V - Cat coop ( ) to V -Mod coop ( ) where V-functors have right adjoints and the object SH op is a bidual for SH. Here we can observe that the right adjoint * D : SH ƒ SH æ æÆ SH of D is the bimodule M above and the composite module SH ƒ D SHƒS ƒ SH op e ƒ SH SH ƒ SH æææÆ SH ƒ SH ƒ SH æææææÆ SH ƒ SH ƒ SH ææ æÆ SH is N. It follows that: a quasi-Hopf structure on the quasialgebra H is precisely a left autonomous structure on the normalized pseudomonoid SH in Therefore Proposition 3.1 applies to yield a Frobenius structure on SH in i coop ( V -Mod ) . coop ( V -Mod ) using E * . This means that SH becomes a pseudomonoid in V -Mod using E * and D * of course, this is not the original algebra structure on H. ; Finally in this example, we can say something about the equivalent definitions of quasi-Hopf algebra: specifically about how p is obtained from a and f . The abovementioned equation satisfied by the element a of H says precisely that it is a (bi)module morphism from E to eo ( 1 ƒ S) o D (note that E is k and eo ( 1 ƒ S) o D is H with appropriate actions). Then p is the pasting composite of the following diagram. 19
Page 1 and 2: Frobenius monads and pseudomonoids
Page 3 and 4: the tensor product), an object I of
Page 5 and 6: qo( koa) qo ( kov) of qo ko( vof) q
Page 7 and 8: only be equivalences rather than is
Page 9 and 10: 2 r :1æÆ æ T satisfying the equa
Page 11 and 12: that m and d are mates as required
Page 13 and 14: Frobenius monad (see pages 151 and
Page 15 and 16: (see [ML] for Yoneda-Day-Kelly inte
Page 17: n: I æÆ æ A ƒA o and counit In
Page 21 and 22: Example Star-autonomous monoidal en
Page 23 and 24: e¢ o ( u¢ e f¢ ) o ( u¢ hf¢ )
Page 25 and 26: that say t is a monad). Notice that
Page 27: [Law1] F.W. Lawvere, Ordinal sums a

Frobenius monads and pseudomonoids Introduction - ResearchGate

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