Note on Frobenius monoidal functors

NOTE ON FR OBENIUS MONOIDA L FUNCTORS BRIAN DA Y AND CRAIG P ASTRO Abstract. It is w ell kno wn that strong monoidal functors preserve duals. In this short no te w e show that a slightly weak er version of functor, whic h we call “F rob enius monoidal”, is sufficient. The idea of this note b ecame apparent from Pro p. 2.8 in the pap er of R. Rose- brugh, N. Sabadini, and R.F.C. W alter s [4]. Throughout suppos e that A and B are strict 1 monoidal categories. Definition 1. A F r ob enius monoidal functor is a functor F : A / / B which is monoidal ( F, r , r 0 ) and co monoidal ( F, i, i 0 ), a nd satisfies the compatibility condi- tions ir = (1 ⊗ r )( i ⊗ 1) : F ( A ⊗ B ) ⊗ F C / / F A ⊗ F ( B ⊗ C ) ir = ( r ⊗ 1)(1 ⊗ i ) : F A ⊗ F ( B ⊗ C ) / / F ( A ⊗ B ) ⊗ F C, for all A, B , C ∈ A . The compact ca se ( ⊗ = ⊕ ) of Co ckett and See ly’s linea rly distributive functors [2] are pr ecisely F rob enius monoidal functors, and F r obenius monoidal functor s with ri = 1 hav e b e en calle d split monoidal b y Szlach´ anyi in [5]. A dual situation in A is a tuple ( A, B , e, n ), where A a nd B are o b jects of A and e : A ⊗ B / / I n : I / / B ⊗ A are morphisms in A , called ev aluation a nd co ev a lua tion resp ectiv ely , satisfying the “triangle iden tities” : A A ⊗ B ⊗ A A 1 ⊗ n / / e ⊗ 1   1 % % L L L L L L L L L L L L L L B B ⊗ A ⊗ B B . n ⊗ 1 / / 1 ⊗ e   1 % % L L L L L L L L L L L L L L Theorem 2. F ro b enius monoidal functors pr eserve dual situations. This theorem is actually a special case of the fact that linear functor s (betw een linear bica teg ories) preser v e linear adjoin ts [1 ]. Date : October 24, 2018. The first author gratefully ac kno wledges partial supp ort of an Australian Researc h Council gran t while the second gratefully ac kno wledges supp ort of an internationa l Macquarie U ni v ers i t y Researc h Scholarship and a Scott Russell Johnson M emor ial Sch olarship. The authors would like to thank Ross Street for several helpful commen ts. 1 W e hav e decided to w ork in the strict setting for simpli cit y of exposition, how ever, this is not necessary . 1 2 BRIAN DA Y AND CRAIG P ASTRO Pr o of. Supp ose that ( A, B , e, n ) is dual situation in A . W e will show that ( F A, F B , e, n ), where e and n ar e defined as e =  F A ⊗ F B F ( A ⊗ B ) r / / F I F e / / I i 0 / /  n =  I F I r 0 / / F ( B ⊗ A ) F n / / F B ⊗ F A i / /  , is a dual situation in B . The follo wing diagr am proves one of the triangle iden tities. F A F A ⊗ F I F A ⊗ F ( B ⊗ A ) F A ⊗ F B ⊗ F A F ( A ⊗ I ) F ( A ⊗ B ⊗ A ) F ( I ⊗ A ) F ( A ⊗ B ) ⊗ F A F I ⊗ F A F A ( † ) 1 ⊗ r 0 / / 1 ⊗ F n / / 1 ⊗ i / / F (1 ⊗ n ) / / i / / i / / 1 $ $ J J J J J J J J J J J 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 ( ( R R R R R R R R R R R R R R R R R r   r   r ⊗ 1   F ( e ⊗ 1)   F e ⊗ 1   i 0 ⊗ 1   The square la belled by ( † ) requir es the second F rob enius condition. W e r e mark that to prov e the other triangle iden tit y is similar and r equires the firs t F robe nius condition.  Prop osition 3. Any s t r ong monoidal functor is a F r ob enius m onoi dal functor. Pr o of. Reca ll that a s trong mono idal functor is a monoida l functor and a comonoidal functor for which r = i − 1 and r 0 = i − 1 0 . The commutativit y of the following dia- gram pr o ves one of the F rob enius conditions. F ( A ⊗ B ) ⊗ F C F A ⊗ F B ⊗ F C F ( A ⊗ B ⊗ C ) F A ⊗ F ( B ⊗ C ) i ⊗ 1 / / i / / r   i O O 1 ⊗ r   1 ⊗ i O O The o ther is similar.  Prop osition 4. The c omp osite of F r ob enius monoida l functors is a F r ob enius monoidal functor. Pr o of. Supp ose that F : A / / B and G : B / / C are F rob enius monoidal func- tors. It is well known a nd easy to see that the compo site of monoidal (r e sp. comonoidal) functors is monoida l (resp. comonoidal). W e therefore need o nly NOTE ON FROB ENIUS MONOIDAL FUNCTORS 3 prov e the F rob enius conditions, one o f which follows from the co mm uta tivity o f GF ( A ⊗ B ) ⊗ GF C G ( F ( A ⊗ B ) ⊗ F C ) GF ( A ⊗ B ⊗ C ) G ( F A ⊗ F B ) ⊗ GF C GF A ⊗ GF B ⊗ GF C G ( F A ⊗ F B ⊗ F C ) GF A ⊗ G ( F B ⊗ F C ) G ( F A ⊗ F ( B ⊗ C )) GF A ⊗ GF ( B ⊗ C ) , ( ‡ ) ($) r / / Gr / / r / / G (1 ⊗ r ) / / 1 ⊗ r / / 1 ⊗ Gr / / Gi ⊗ 1   G ( i ⊗ 1)   i ⊗ 1   i   Gi   i   where the squar e lab elled b y ( ‡ ) uses the F r obenius prop ert y o f F , and the squar e lab elled by ($) uses the F r obenius prop ert y of G . The other F r obenius condition follo ws from a similar diagra m.  It is not too difficult to see that a F rob enius monoidal functor F : 1 / / A is a F rob enius a lgebra in A . Therefore, we have the following coro lla ry . Corollary 5. F r ob enius monoidal funct ors pr eserve F r ob en ius algebr as. That is, if R is a F r ob enius algebr a in A and F : A / / B is a F r ob eniu s functor, then F R is a F r ob enius algebr a in B . Example 6. Supp ose that A is a braided mono idal category . If R ∈ A is a F rob enius algebra in A , then F = R ⊗ − : A / / A is a F rob enius monoidal functor. The monoidal structure ( F, r, r 0 ) is given by r A,B =  R ⊗ A ⊗ R ⊗ B R ⊗ R ⊗ A ⊗ B 1 ⊗ c ⊗ 1 / / R ⊗ A ⊗ B µ ⊗ 1 ⊗ 1 / /  r 0 =  I R η / /  and the comonoida l str ucture ( F , i, i 0 ) b y i A,B =  R ⊗ A ⊗ B R ⊗ R ⊗ A ⊗ B δ ⊗ 1 ⊗ 1 / / R ⊗ A ⊗ R ⊗ B 1 ⊗ c ⊗ 1 / /  i 0 =  R I ǫ / /  . The F rob enius conditions now follow easily fro m the prop erties of F rob enius alge- bras. This example shows that F rob enius mono idal functors g eneralize F r obenius al- gebras muc h in the same way that mo noidal comonads , o r co mo noidal monads, generalize bialgebras. The following pro position is a genera lization of the fact that morphisms of F rob e- nius algebra s (mor phisms whic h are b oth algebra and coa lgebra morphisms) are isomorphisms. It also genera liz es the res ult that monoidal natural tra nsformations betw een str o ng monoidal functors w ith (left o r rig h t) compa ct domain ar e in vert- ible. Prop osition 7. Supp ose t hat F, G : A / / B ar e F r ob enius monoidal funct ors and that α : F / / G is a monoidal and c omonoida l natur al tr ansformation. If A ∈ A is p art of a dual s it u ation, i.e., ( A, B , e, n ) or ( B , A, e, n ) is a dual situation, then α A : F A / / GA is invertible. 4 BRIAN DA Y AND CRAIG P ASTRO Pr o of. W e sha ll assume that A is par t of the dual situation ( A, B , e, n ). The other case is treated similarly . The co mponent α B : F B / / GB has mate GA GA ⊗ F B ⊗ F A 1 ⊗ n / / GA ⊗ GB ⊗ F A 1 ⊗ α B ⊗ 1 / / F A e ⊗ 1 / / which we will show is the inv erse to α A . If α is b o th monoida l and comonoidal then the dia grams F A ⊗ F B F ( A ⊗ B ) F I I GA ⊗ GB G ( A ⊗ B ) GI r   F e   r   Ge   α A ⊗ α B / / α A ⊗ B / / α I / / i 0 # # F F F F F F F F F i 0 { { x x x x x x x x x F B ⊗ F A F ( B ⊗ A ) F I I GB ⊗ GA G ( B ⊗ A ) GI i   F n   i   Gn   α A ⊗ α B / / α A ⊗ B / / α I / / r 0 { { x x x x x x x x x r 0 # # F F F F F F F F F commute. The following diagra ms pr o ve that α A is inv ertible.The first diagram ab o ve says exactly that the triang le la b elled by ( £ ) b elow co mm utes. The second diagram abov e that the triangle lab elled by ( U ) below co mm utes. F A GA F A ⊗ F B ⊗ F A F A GA ⊗ F B ⊗ F A GA ⊗ GB ⊗ F A ( £ ) α / / α ⊗ 1 ⊗ 1 / / e ⊗ 1 o o 1 ⊗ n   e ⊗ 1   1 ⊗ n   1 ⊗ α ⊗ 1   α ⊗ α ⊗ 1 ( ( R R R R R R R R R R R R R R R R GA GA ⊗ F B ⊗ F A GA ⊗ GB ⊗ GA GA GA ⊗ GB ⊗ F A F A ( U ) 1 ⊗ n / / 1 ⊗ 1 ⊗ α o o α o o 1 ⊗ α ⊗ α v v l l l l l l l l l l l l l l l l 1 ⊗ n   e ⊗ 1   1 ⊗ α ⊗ 1   e ⊗ 1    Denote by F ro b ( A , B ) the category of F r obenius mono idal functors from A to B and all natural tra nsformations b et ween them. Prop osition 8 (cf. [4] Prop. 2.10) . If B is a br aide d monidal c ate gory, then F rob( A , B ) is a br aide d monoidal c ate gory with the p ointwise t ensor pr o duct of functors. Pr o of. Co nsider the p oint wise tenso r pro duct of F rob enius monoida l functors F , G : A / / B . That is, ( F ⊗ G ) A = F A ⊗ GA. It is obviously a n asso ciativ e a nd unital tensor pro duct with unit I ( A ) = I for all A ∈ A . NOTE ON FROB ENIUS MONOIDAL FUNCTORS 5 W e ma y define morphisms as follo ws: r = ( r ⊗ r )(1 ⊗ c − 1 ⊗ 1) : ( F ⊗ G ) A ⊗ ( F ⊗ G ) B / / ( F ⊗ G )( A ⊗ B ) r 0 = r 0 ⊗ r 0 : I / / ( F ⊗ G ) I i = (1 ⊗ c ⊗ 1)( i ⊗ i ) : ( F ⊗ G )( A ⊗ B ) / / ( F ⊗ G ) A ⊗ ( F ⊗ G ) B i 0 = i 0 ⊗ i 0 : ( F ⊗ G ) I / / I . That these morphisms pr ovide a monoidal and a como no idal structure o n F ⊗ G is no t to o difficult to show, and is omitted here. T he following dia gram proves the first F rob enius co nditio n, wher e the ⊗ sy mbo l has b een remov ed as a space spac ing mechanism. F ( AB ) G ( AB ) F C GC F A F B GA GB F C GC F A GA F B GB F C GC F ( AB ) F C G ( AB ) GC F ( AB C ) G ( AB C ) F A F B F C GA GB GC F A F ( B C ) GA G ( B C ) F A GA F B F C GB GC F A GA F ( B C ) G ( B C ) ( F ⊗ G )( AB ) ⊗ ( F ⊗ G ) C ( F ⊗ G ) A ⊗ ( F ⊗ G )( BC ) i i 1 1 / / 1 c 1 1 1 / / i 1 i 1 / / 1 c F BF C ,GA 1 1 / / i i / / 1 c 1 / / 1 c − 1 1   1 1 c − 1 GAGB,F C 1   1 1 1 c − 1 1   r r   1 r 1 r   1 1 r r   The b ottom left square commutes by the F rob enius condition, and the o ther s b y prop erties of the br aiding. The second F rob enius condition follows from a similar diagram. So , F ⊗ G is a F rob enius monoidal functor. The braiding c F, G : F ⊗ G / / G ⊗ F is given on components by ( c F, G ) A = c F A,GA : F A ⊗ GA / / GA ⊗ F A.  Corollary 9. If B is a br aide d monoidal c ate gory and A is a s elf-dual c omp act c ate gory, me aning that for any obje ct A ∈ A , ( A, A, e, n ) is a dual situation in A , then F rob( A , B ) is a self-dual br aide d c omp act c ate gory. Pr o of. B y Theorem 2 F rob e nius monoidal functors preserve duals, and therefore, for an y A ∈ A , ( F A, F A, e, n ) is a dual situation in B .  Recall that, if A is a sma ll monoidal categ ory , and if small colimits exist and commute with the tensor pr oduct in B , then the equa tions F ∗ G = Z A,B A ( A ⊗ B , − ) · F A ⊗ F B J = A ( I , − ) · I , where · denotes co pow er, describ e the c onvolution monoidal stru ctur e on the functor category [ A , B ] (cf. [3]). Then w e ha ve: Theorem 10. If A is a s m al l monoidal c ate gory and B is a monoidal c ate gory hav- ing all smal l c olimits c ommuting with ten sor, then any F r ob enius monoida l functor 6 BRIAN DA Y AND CRAIG P ASTRO F : A / / B fo r which the c anonic al evaluation m orphism ( ♭ ) Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) / / F is an isomorphism, b e c omes an algeb r a with a c omultiplic ation which satisfies the F r ob enius identities in the c onvolution functor c ate gory [ A , B ] . Note that, by t he Y one da lemma, the e quation ( ♭ ) is satisfie d by al l the fun ctors F : A / / B if A is a close d monoi dal c ate gory and the c anonic al evaluation morphism ( ♯ ) Z B ,C A ( A, B ⊗ C ⊗ [ B ⊗ C, − ]) / / A ( A, − ) is an isomorphism for al l A ∈ A . Before w e prove Theor em 10 w e will need the following lemma. Lemma 11. Assuming e quation ( ♭ ) in The or em 10, we m ay also derive the two variable version, that is, that the c anonic al evaluation morphism Z A,B A ( A ⊗ B , − ) · F ( A ⊗ B ) / / F is an isomorphism. Pr o of. The canonical ev aluation morphism Z A,B ,C A ( A ⊗ B ⊗ C , − ) · F ( A ⊗ B ⊗ C ) Z A,B A ( A ⊗ B , − ) · F ( A ⊗ B ) h / / is a retra ction o f (either of the cano nical mo rphisms in the opp osite direction), say , k . W e may comp ose the canonical morphism Z A,B A ( A ⊗ B , − ) · F ( A ⊗ B ) / / F with the isomo r phism F Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) ∼ = / / of o ur assumption to g et a morphism Z A,B A ( A ⊗ B , − ) · F ( A ⊗ B ) Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) l / / , NOTE ON FROB ENIUS MONOIDAL FUNCTORS 7 which ma k es the diagram A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) Z A,B A ( A ⊗ B , − ) · F ( A ⊗ B ) Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) copr 4 4 i i i i i i i i i i i i i i i i i i i i i i i copr / / copr * * U U U U U U U U U U U U U U U U U U U U U U U h   l   commute. Therefore lh = 1. W e ha v e hl = hl hk = hk = 1 so l is an isomorphism, henc e the canonical ev aluation morphism Z A,B A ( A ⊗ B , − ) · F ( A ⊗ B ) / / F is a n isomorphism.  A consequence o f Lemma 11 is that we may wr ite F ∗ F = Z X,C A ( X ⊗ C, − ) · F X ⊗ F C ∼ = Z X,C A ( X ⊗ C, − ) ·  Z A,B A ( A ⊗ B , X ) · F ( A ⊗ B )  ⊗ F C ∼ = Z X,A,B ,C ( A ( X ⊗ C, − ) × A ( A ⊗ B , X )) · ( F ( A ⊗ B ) ⊗ F C ) ∼ = Z A,B ,C  Z X A ( X ⊗ C, − ) × A ( A ⊗ B , X )  · ( F ( A ⊗ B ) ⊗ F C ) ∼ = Z A,B ,C A ( A ⊗ B ⊗ C , − ) · F ( A ⊗ B ) ⊗ F C, (Y oneda) and s imila rly , F ∗ F ∼ = Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F A ⊗ F ( B ⊗ C ) . Pr o of of The or em 10. Using the isomorphisms o f equation ( ♭ ) and Lemma 11 one of the F rob enius equations may b e written a s Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ) ⊗ F C Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F A ⊗ F B ⊗ F C Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F A ⊗ F ( B ⊗ C ) . R 1 ⊗ i ⊗ 1 / / R 1 ⊗ i / / R 1 ⊗ r   R 1 ⊗ 1 ⊗ r   8 BRIAN DA Y AND CRAIG P ASTRO This diagr am is se e n to comm ute a s F is a F rob enius monoidal functor. The other F rob enius eq uation follows fro m a simila r diagr am. T o pr o ve the s econd par t of the theo r em, a ssume that A is a clo sed monoida l category a nd that equation ( ♯ ) holds. The following calculation verifies the claim. Z A,B ,C A ( A ⊗ B ⊗ C, − ) · F ( A ⊗ B ⊗ C ) ∼ = Z A,B ,C A ( C, [ A ⊗ B , − ]) · F ( A ⊗ B ⊗ C ) ( A closed) ∼ = Z A,B F ( A ⊗ B ⊗ [ A ⊗ B , − ]) (Y oneda) ∼ = Z X,A,B A ( X , A ⊗ B ⊗ [ A ⊗ B , − ]) · F X (Y oneda) ∼ = Z X A ( X , − ) ⊗ F X ( ♯ ) ∼ = F (Y oneda)  References [1] J. R. B. Co c ke tt, J. Koslowski and R. A . G. Seely . Introduction to linear bicategories, Math. Struct. Comp. Science 10 no. 2 (2000): 165–203. [2] J. R. B. Co c k ett and R. A. G. Seely . Linearly distributive functors, J. P ur e Appl. Algebra 143 (1999): 155–203. [3] Brian Da y . On closed categories of f unct ors, in R ep orts of the Midwest Cate gory Seminar IV , Lecture Notes i n M athematics 137 (1970): 1–38. [4] R. Rosebrugh, N. Sabadini and R.F.C. W alters. Generic commutat ive separable algebras and cospans of graphs, Theory Appl. Categories 15 (2005): 164–177. [5] Korn ´ el Szlach´ an yi. Fi nite quan tum group oids and inclusions of finite type, Fields Inst. Comm. 30 (2001): 393–407. Dep ar tment of M a thema tics, Macquarie Un iversity, New South W ales 2109 A us- tralia E-mail addr ess : craig@ic s.mq.edu.a u