On endomorphism algebras of separable monoidal functors

ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTORS BRIAN DA Y AND CRAIG P ASTRO Abstract. W e show that the (co) endomorphism algebra of a sufficien tly sep- arable “fibre” functor into V ect k , for k a field of charact eristic 0, has the structure of what we call a “unital” von Neumann core in V ect k . F or V e ct k , this particular notion of algebra is we aker than that of a Hopf algebra, al though the corresp onding concept in Set is again that of a group. 1. Introduction Let ( C , ⊗ , I , c ) be a symmetric (or just bra ided) monoidal ca tegory . Recall that an algebr a in C is a n ob ject A ∈ C equipp ed with a multiplication µ : A ⊗ A / / A and a unit η : I / / A satisfying µ 3 = µ (1 ⊗ µ ) = µ ( µ ⊗ 1) : A ⊗ 3 / / A (asso ciativity) a nd µ ( η ⊗ 1 ) = 1 = µ (1 ⊗ η ) : A / / A (unit conditions). Dually , a c o algebr a in C is a n ob ject C ∈ C equipped with a comultiplication δ : C / / C ⊗ C and a counit ǫ : C / / I sa tisfying δ 3 = (1 ⊗ δ ) δ = ( δ ⊗ 1 ) δ : C / / C ⊗ 3 (coasso ciativity) and ( ǫ ⊗ 1 ) δ = 1 = (1 ⊗ ǫ ) δ : C / / C (counit conditions). A very we ak bialgebr a in C is an o b ject A ∈ C with b oth the structure of a n algebra a nd a coalg ebra in C related by the axiom δ µ = ( µ ⊗ µ )(1 ⊗ c ⊗ 1)( δ ⊗ δ ) : A ⊗ A / / A ⊗ A. F or ex ample, any k -bialgebr a or weak k - bialgebra is a very w ea k bialgebr a in this sense (for C = V ect k ). The str ucture A is then called a von Neumann c or e in C if it als o has an ant ipo de S : A / / A satisfying the axiom µ 3 (1 ⊗ S ⊗ 1 ) δ 3 = 1 : A / / A. F or example, the set of all finite paths of edges in a (r o w-finite) gr aph algebra [8] forms a von Neumann cor e in C = Set , and so do es any gro up in Set . Since groups A in Set are characteriz ed by the (stro nger) axio m ( † ) 1 ⊗ η = (1 ⊗ µ )(1 ⊗ S ⊗ 1 ) δ 3 : A / / A ⊗ A, a very weak bialgebra A satisfying ( † ), in the gener al C , will b e called a unital von Neumann core in C . Such a unital cor e A always has a left in verse, namely (1 ⊗ µ )(1 ⊗ S ⊗ 1)( δ ⊗ 1), to the “ fusion” o per a tor (1 ⊗ µ )( δ ⊗ 1) : A ⊗ A / / A ⊗ A, and the latter satisfies the fusion equation [9]. An y Ho pf algebra in C satisfies the axiom ( † ), and in this a rticle we a re mainly interested in pro ducing a unital von Date : No vem b er 21, 2018. The fir st author gratefully ackno wledges partial s upp ort of an Australi an Researc h Council gran t while the second gratefully ackno wledges support of an international Macquarie Unive r sit y Researc h Sc holar ship. 1 2 BRIAN D A Y AND CRAIG P ASTRO Neumann co re, na mely E nd ∨ U , a sso ciated to a certain type of monoida l functor U int o V ect k . How ever, it will not b e the case that all unital von Neumann cores in V ect k can be repro duced as suc h. W e will tacitly a ssume throughout the a rticle that the g round category [7] is V ect = V ect k , for k a field of characteristic 0, so that the categ o ries and functors considered here are all k - linear (although a ny reas onable categ ory [ D , V ect ] o f pa - rameterized vector spaces would suffice). W e denote by V ect f the full sub categor y of V ect consis ting of the finite dimensiona l v ector spaces , a nd we further s upp ose that ( C , ⊗ , I , c ) is a braided monoidal category with a “fibre” functor U : C / / V ect which has b oth a monoidal str ucture ( U, r, r 0 ) and a comonoidal structure ( U, i, i 0 ). W e call U sep ar able 1 if r i = 1 and i 0 r 0 = dim( U I ) · 1; i.e., for all A, B ∈ C , the diagrams U ( A ⊗ B ) U A ⊗ U B U ( A ⊗ B ) i / / r   1 % % L L L L L L L L L L L I U I I r 0 / / i 0   dim U I · 1 # # G G G G G G G G G G G G commute. First we pr oduce an algebra structure ( µ, η ) on End ∨ U = Z C U C ∗ ⊗ U C using the monoidal and comonoidal str uctures on U . Secondly , we supp ose that C has a suitable small genera ting set A of ob jects, a nd pr o duce a coa lgebra str ucture ( δ, ǫ ) on End ∨ U when each v alue U A , A ∈ A , is finite dimensional. Finally , we assume that U is equipped with a natural non-degenerate form U ( A ∗ ) ⊗ U A / / k suitably rela ted to the ev aluatio n and co ev aluation maps of C and V ect f , where each A ∈ A has a ⊗ - dual A ∗ in C whic h again lies in A . This last as sumption is sufficient to provide End ∨ U with an ant ip o de s o that it beco mes a unital von Neumann core in the ab ov e sense . By wa y of examples, we note that many separ able monoida l functors are con- structable from separ able monoidal categor ies; i.e ., from mo no idal ca tegories C for which the tensor pro duct map ⊗ : C ( A, B ) ⊗ C ( C, D ) / / C ( A ⊗ C, B ⊗ D ) is a natur a lly split epimorphism (as is the cas e fo r s ome finite car tesian pr oducts such as V ect n f ). A clo sely r elated so urce of ex a mples is the no tion of a weak dimension functor on C (cf. [5]); this is a como no idal functor ( d, i, i 0 ) : C / / Set f 1 Strictly , w e s hould also require the conditions ( r ⊗ 1)(1 ⊗ i ) = ir : U A ⊗ U ( B ⊗ C ) / / U ( A ⊗ B ) ⊗ U C, and (1 ⊗ r )( i ⊗ 1) = ir : U ( A ⊗ B ) ⊗ U C / / U A ⊗ U ( B ⊗ C ) in order for U to b e called “separable”, but we do not need these here. ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 3 for whic h the comonoidal tra nsformation co mponents i = i C,D : d ( C ⊗ D ) / / dC × dD are injective functions, while the unique map i 0 : dI / / 1 is surjective. V arious examples are descr ibed at the conclusion o f the pap er. W e supp ose the rea der is familiar to so me extent with the standard references on the pro blem when restricted to the cas e of U s trong monoida l. W e would like to thank Ross Street for several helpful comments. 2. The algebraic structure on E nd ∨ U If C is a ( k -linea r) mono idal ca tegory and U : C / / V ect has a monoidal structure ( U , r, r 0 ) and a comonoidal s tructure ( U, i, i 0 ), then End ∨ U has an as s oc ia tiv e a nd unital k -alge bra s tr ucture who se multiplication µ is the comp osite ma p Z C U C ∗ ⊗ U C ⊗ Z D U D ∗ ⊗ U D Z C,D U C ∗ ⊗ U D ∗ ⊗ U C ⊗ U D ∼ =   Z C,D ( U C ⊗ U D ) ∗ ⊗ U C ⊗ U D can   Z C,D U ( C ⊗ D ) ∗ ⊗ U ( C ⊗ D ) R i ∗ ⊗ r / / Z B U B ∗ ⊗ U B R ⊗ O O µ / / while the unit η is given by k k ∗ ⊗ k ∼ =   U I ∗ ⊗ U I . i ∗ 0 ⊗ r 0 / / Z C U C ∗ ⊗ U C copr C = I O O η / / The asso ciativity and unit axioms for (End ∨ U, µ, η ) now follow directly fr o m the corres p onding as so ciativit y and unit ax ioms for ( U, r, r 0 ) and ( U, i, i 0 ). An augmen- tation ǫ is given by U D ∗ ⊗ U D Z C U C ∗ ⊗ U C copr C = D O O k ǫ / / ev : : u u u u u u u u u u u u u u in V e ct , where ǫ η = dim U I · 1 . W e also observe that the co end End ∨ U = Z C U C ∗ ⊗ U C 4 BRIAN D A Y AND CRAIG P ASTRO actually exists in V ect if C co n ta ins a s mall full sub catego ry A with the pro perty that the family { U f : U A / / U C | f ∈ C ( A, C ) , A ∈ A } is epimorphic in V e ct for each o b ject C ∈ C . In fact, we s hall use the stro nger condition that the maps α C : Z A ∈ A C ( A, C ) ⊗ U A / / U C should b e isomo rphisms, not just epimorphisms . This stro ng er co ndition implies that we can effectively r e place R C ∈ C by R A ∈ A since Z C U C ∗ ⊗ U C ∼ = Z C U C ∗ ⊗ ( Z A C ( A, C ) ⊗ U A ) ∼ = Z A U A ∗ ⊗ U A by the Y oneda lemma. If we furthermore a sk that each v alue U A b e finite dimensio nal for A in A , then End ∨ U ∼ = Z A ∈ A U A ∗ ⊗ U A is canonica lly a k -coa lgebra with co unit the augmentation ǫ , and comultiplication δ given by U A ∗ ⊗ U A Z A U A ∗ ⊗ U A copr O O Z A U A ∗ ⊗ U A ⊗ Z A U A ∗ ⊗ U A δ / / U A ∗ ⊗ U A ⊗ U A ∗ ⊗ U A, 1 ⊗ n ⊗ 1 / / copr ⊗ cop r O O where n denotes co ev aluation in V ec t f . Prop osition 2.1 . If U is sep ar able then End ∨ U satisfies t he k -bialgebr a axiom End ∨ U ⊗ End ∨ U End ∨ U µ   End ∨ U ⊗ End ∨ U. δ / / (End ∨ U ) ⊗ 4 δ ⊗ δ / / (End ∨ U ) ⊗ 4 1 ⊗ c ⊗ 1   µ ⊗ µ   Pr o of. Let B denote the mono idal full sub category of C generated by A (w e will essentially replace C by this small category B ). Then, for all C , D in B , w e have, by induction on the tensor lengths of C and D , that U ( C ⊗ D ) is finite dimensio nal since it is a retract of U C ⊗ U D . Moreov e r , we have Z A ∈ A U A ∗ ⊗ U A ∼ = Z B ∈ B U B ∗ ⊗ U B ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 5 by the Y oneda lemma, since the natural family α B : Z A ∈ A C ( A, B ) ⊗ U A / / U B is a n isomo rphism for all B ∈ B . Since ri = 1, the triangle k ( U C ⊗ U D ) ⊗ ( U C ⊗ U D ) ∗ U ( C ⊗ D ) ⊗ U ( C ⊗ D ) ∗ r ⊗ i ∗   n 6 6 n n n n n n n n n n ) ) R R R R R R R R R R commutes in V ect f , where n denotes the co ev aluation maps. The asser ted bia lgebra axiom then holds on End ∨ U s ince it reduces to the following diag ram on filling in the definitions of µ and δ (where , for the moment, we have dropp ed the sy m b ol “ ⊗ ”): U C U C ∗ U D U D ∗ U C U D U C ∗ U D ∗ U C U D ( U C U D ) ∗ U ( C D ) U ( C D ) ∗ U C ( U C U C ∗ ) U C ∗ U D ( U D U D ∗ ) U D ∗ U C U D U C U D U C ∗ U D ∗ U C ∗ U D ∗ U C U D U C U D ( U C U D ) ∗ ( U C U D ) ∗ U ( C D ) U ( C D ) U ( C D ) ∗ U ( C D ) ∗ ∼ =   ∼ =   r i ∗   ∼ =   ∼ =   r r i ∗ i ∗   1 n 1 1 n 1 / / 1 n 1 / / 1 n 1 / / for all C, D ∈ B .  Notably the bialg ebra ax io m End ∨ U ⊗ End ∨ U End ∨ U µ / / k ǫ            ǫ ⊗ ǫ   ? ? ? ? ? ? ? ? ? do es not hold in general, while the form of the a xiom k End ∨ U η _ _ ? ? ? ? ? ? ? ? ? End ∨ U ⊗ End ∨ U δ / / η ⊗ η ? ?          holds where we multiply δ by dim U I . The k - bialgebra axio m established in the above pr opo sition implies that the “fusion” o p era tor (1 ⊗ µ )( δ ⊗ 1) : A ⊗ A / / A ⊗ A s a tisfies the fusion equation (see [9 ] for details). 6 BRIAN D A Y AND CRAIG P ASTRO The k -linear dual of End ∨ U is of course [ Z C U C ∗ ⊗ U C , k ] ∼ = Z C [ U C ∗ , U C ∗ ] which is the endomor phism k -algebra of the functor U ( − ) ∗ : C op / / V ect . If ob A is finite, so that Z A U A ∗ ⊗ U A is finite dimensiona l, then Z C [ U C ∗ , U C ∗ ] ∼ = Z A [ U A ∗ , U A ∗ ] is a lso a k -coa lgebra. 3. The un it al von Neumann antipode W e now take ( C , ⊗ , I , c ) to b e a braided monoidal categ ory and A ⊂ C to be a small full sub catego ry o f C for which the monoidal and comonoida l functor U : C / / V ect induces U : A / / V ect f on r estriction to A . W e supp ose that A is s uch that • the identit y I o f ⊗ lies in A , a nd each ob ject of A ∈ A has a ⊗ -dual A ∗ lying in A . With respect to U , we suppos e A has the pro per ties • “ U -ir reducibilit y”: A ( A, B ) 6 = 0 implies dim U A = dim U B for a ll A, B ∈ A , • “ U -dens ity”: the canonical map α C : Z A ∈ A C ( A, C ) ⊗ U A / / U C is an is o morphism for all C ∈ C , • “ U -tr ace”: ea c h ob ject of A has a U -trace in C ( I , I ), wher e b y U -trace of A ∈ A w e mean an iso morphism d ( A ) in C ( I , I ) such that the following t wo diagrams commu te. I A ⊗ A ∗ n   A ∗ ⊗ A c / / I e O O d ( A ) / / k U I r 0   U I dim U I · U ( d ( A )) / / k dim U A / / r 0   W e shall a ssume dim U I 6 = 0 s o that the latter assumption implies dim U A 6 = 0, for all A ∈ A . W e requir e also a na tural is omorphism u = u A : U ( A ∗ ) U A ∗ ∼ = / / ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 7 such tha t ( n, r, r 0 ) k U I r 0 / / U ( A ⊗ A ∗ ) U n   U A ⊗ U A ∗ n   U A ⊗ U ( A ∗ ) 1 ⊗ u − 1   ? ? ? ? ? ? r ? ?       commutes, and ( e, i , i 0 ) U ( A ∗ ⊗ A ) U I U e O O k i 0 / / U ( A ∗ ) ⊗ U A i   ? ? ? ? ? ? U A ∗ ⊗ U A u ⊗ 1 ? ?       e O O commutes. This means that U “pres erves duals” when restricted to A . An endomorphism σ : End ∨ U / / End ∨ U may b e defined b y comp onents U A ∗ ⊗ U A U ( A ∗ ) ∗ ⊗ U ( A ∗ ) , σ A / / Z A U A ∗ ⊗ U A copr O O Z A U A ∗ ⊗ U A copr O O σ / / each σ A being given by commutativit y of U A ∗ ⊗ U A U ( A ∗ ) ∗ ⊗ U ( A ∗ ) σ A / / U A ∗ ⊗ U A ∗∗ 1 ⊗ ρ   U ( A ∗ ) ⊗ U ( A ∗ ) ∗ u − 1 ⊗ u ∗ / / c O O where ρ denotes the canonical isomorphism from a finite dimensional v ector space to its double dual. Clearly each comp onent σ A is invertible. Theorem 3. 1. L et C , A , and U b e as ab ove, and supp ose that U is br aide d and sep ar able as a monoida l functor. Then ther e is an invertible antip o de S on End ∨ U such that (E nd ∨ U, µ, η , δ, ǫ, S ) is a un ital von Neum ann c or e in V ect k . Pr o of. A family of ma ps { S A | A ∈ A } is defined b y S A = dim U I · (dim U A ) − 1 · σ A . 8 BRIAN D A Y AND CRAIG P ASTRO Then, by the U -irr educibilit y assumption on the catego ry A , this family induces an invertible endomorphism S o n the co end End ∨ U ∼ = ∞ X n =1 Z A ∈ A n U A ∗ ⊗ U A, where A n is the full sub category of A determined b y { A | dim U A = n } . W e no w take S to b e the a ntipo de on End ∨ U and chec k that 1 ⊗ η = (1 ⊗ µ )(1 ⊗ S ⊗ 1) δ 3 . F ro m the definition of µ and δ , we r equire commutativit y of the exterior of the following dia gram (where, ag ain, we hav e dropp ed the symbol “ ⊗ ”): U A ∗ U A U A ∗ U A U A ∗ U A U A ∗ U A U A ∗ U A U A ∗ U A U A ∗ U A I U A ∗ U A U ( A ∗ ) ∗ U ( A ∗ ) U A ∗ U A U A ∗ U A U A ∗∗ U A ∗ U A ∗ U A ∼ =   U A ∗ U A U ( A ∗ ) ∗ U ( A ∗ ) U A ∗ U A ∼ =   U A ∗ U A ( U ( A ∗ ) U A ) ∗ U ( A ∗ ) U A ∼ =   U A ∗ U A U ( A ∗ A ) ∗ U ( A ∗ A ) 1 1 i ∗ r   U A ∗ U A Z B U B ∗ U B 1 1 copr   U A ∗ U A U A U A ∗ U A ∗ U A 1 1 n 1 1 5 5 l l l l l l l l l l l l l U A ∗ U A U A U A ∗ U A ∗ U A 1 1 1 c 1 O O 1 1 c 1 1 i i R R R R R R R R R R R R R 1 n 1 1 1 O O 1 n 1 O O ∼ = O O 1 1 S A 1 1 / / 1 1 e ∗ 1 1 / / 1 1 η / / (1) (2) (3) The r egion labelled by (1) comm utes on comp osition with 1 ⊗ n ⊗ 1 since k U A ⊗ U A ∗ n / / U A ⊗ U A ⊗ U A ∗ ⊗ U A ∗ 1 ⊗ n ⊗ 1   U A ⊗ U A ⊗ U A ∗ ⊗ U A ∗ 1 ⊗ 1 ⊗ c   U A ⊗ U A ∗ ⊗ U A ⊗ U A ∗ 1 ⊗ c ⊗ 1   U A ⊗ U A ∗ n   n ⊗ 1 ⊗ 1 / / ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 9 commutes (choose a basis for U A ). The reg ion lab elled by (2) now commutes by insp e ction of: U A ⊗ U A ∗ U A ⊗ U A ⊗ U A ∗ ⊗ U A ∗ 1 ⊗ n ⊗ 1 O O U A ⊗ U A ⊗ U A ∗ ⊗ U A ∗ 1 ⊗ 1 ⊗ c O O U A ⊗ U A ∗ ⊗ U A ⊗ U A ∗ 1 ⊗ c ⊗ 1 O O U A ⊗ U A ∗∗ ⊗ U A ∗ ⊗ U A ∗ 1 ⊗ e ∗ ⊗ 1 / / U A ⊗ U A ∗∗ ⊗ U A ∗ ⊗ U A ∗ 1 ⊗ 1 ⊗ c O O U A ⊗ U A ∗ ⊗ U A ∗∗ ⊗ U A ∗ 1 ⊗ c ⊗ 1 O O U A ⊗ U ( A ∗ ) ⊗ U ( A ∗ ) ∗ ⊗ U A ∗ O O U A ⊗ U ( A ∗ ) ∗ ⊗ U ( A ∗ ) ⊗ U A ∗ O O 1 ⊗ u − 1 ⊗ u ∗ ⊗ 1 O O 1 ⊗ c ⊗ 1 O O 1 ⊗ σ A ⊗ 1 / / 1 ⊗ 1 ⊗ ρ ⊗ 1 R R R R R R R R R R R R R R R ) ) R R R R R R R R R R R R R R R 1 ⊗ ρ ⊗ 1 ⊗ 1 W W W W W W W W W W W + + W W W W W W W W W W W where the top leg of (2) ha s b een rescaled b y a factor o f (dim U I ) − 1 · dim U A . F ro m the definition of the U - trace d ( A ) of A ∈ A , we hav e that k k dim U I · (dim U A ) − 1 / / U I r 0   U I r 0   U ( d ( A ) − 1 ) / / commutes, so that the exter io r of k k dim U I · (dim U A ) − 1 O O U A ⊗ U A ∗ n O O U A ⊗ U A ∗ 1 ⊗ u − 1 : : t t t t t t t t U ( A ⊗ A ∗ ) r $ $ J J J J J J J J U I r 0 / / U I U ( d ( A ) − 1 ) O O U n O O r 0 / / ( n,r,r 0 ) commutes. 10 BRIAN D A Y AND CRAIG P ASTRO Thu s the region lab elled b y (3), with the to p leg rescaled by the facto r dim U I · (dim U A ) − 1 , co mm utes on examination of the following diag ram: k ∗ ⊗ k k ∗ ⊗ U A ⊗ U A ∗ k ∗ ⊗ U A ∗ ⊗ U A ( U A ∗ ⊗ U A ) ∗ ⊗ U A ∗ ⊗ U A ( U ( A ∗ ) ⊗ U A ) ∗ ⊗ U ( A ∗ ) ⊗ U A U ( A ∗ ⊗ A ) ∗ ⊗ U ( A ∗ ⊗ A ) Z B U B ∗ ⊗ U B k ∗ ⊗ U ( A ∗ ) ⊗ U A 1 ⊗ u − 1 ⊗ 1 M M M & & M M M U I ∗ ⊗ U ( A ∗ ⊗ A ) i ∗ 0 ⊗ r M M M & & M M M U e ∗ ⊗ 1 ) ) T T T T T T T T T T T T k ∗ ⊗ U A ⊗ U ( A ∗ ) 1 ⊗ 1 ⊗ u − 1 M M M & & M M M U I ∗ ⊗ U ( A ⊗ A ∗ ) i ∗ 0 ⊗ r M M M & & M M M U I ∗ ⊗ U I U I ∗ ⊗ U I 1 9 9 r r r r r r r r r r r r copr   4 4 4 4 4 4 i ∗ 0 ⊗ r / / copr / / 1 ⊗ dim U I · (dim U A ) − 1 · n O O 1 ⊗ c O O e ∗ ⊗ 1 ⊗ 1 4 4 i i i i i i i i i i i i i i i ( u ⊗ 1) ∗ ⊗ ( u − 1 ⊗ 1) * * U U U U U U U U U U U U U U i ∗ ⊗ r   copr   1 ⊗ c O O 1 ⊗ U c O O 1 ⊗ U e   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 ⊗ U ( d ( A ) − 1 ) · U n O O ( n,r,r 0 ) ( ∗ ) ( e,i,i 0 ) whose commutativit y dep ends on the hypothesis that ( U, r, r 0 ) is braided monoidal in or der fo r U A ⊗ U ( A ∗ ) U ( A ∗ ) ⊗ U A c / / U ( A ∗ ⊗ A ) r   U ( A ⊗ A ∗ ) r   U c / / ( ∗ ) to co mm ute.  4. The fusion opera tor Let E = E nd ∨ U . The unital von Neuma nn axiom on E implies that the fusion op erator f = (1 ⊗ µ )( δ ⊗ 1 ) : E ⊗ E / / E ⊗ E ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 11 has a le ft inv erse, namely g = (1 ⊗ µ )(1 ⊗ S ⊗ 1 )( δ ⊗ 1). F o r this we consider the following dia gram: E ⊗ E E ⊗ 3 E ⊗ E E ⊗ 4 E ⊗ 3 E ⊗ 4 E ⊗ 3 E ⊗ E E ⊗ 3 . δ ⊗ 1 / / 1 ⊗ η ⊗ 1   δ 3 ⊗ 1 % % L L L L L L L L L L L L 1   1 ⊗ µ / / 1 ⊗ δ ⊗ 1   δ ⊗ 1 ⊗ 1   δ ⊗ 1   1 ⊗ 1 ⊗ µ * * U U U U U U U U U U 1 ⊗ S ⊗ 1 ⊗ 1   1 ⊗ S ⊗ 1   1 ⊗ µ   1 ⊗ µ ⊗ 1 o o 1 ⊗ 1 ⊗ µ % % L L L L L L L L L L L L 1 ⊗ µ o o In particular f = (1 ⊗ µ )( δ ⊗ 1 ) is a partial isomorphism, i.e., f g f = f and g f g = g . 5. Examples of sep arable monoidal functors in the present context Unless otherwis e indicated, categories, functor s, and natural transfor mations shall be k -linear, for k a suitable field. F or these exa mples we recall that a (small) k -linear pr omonoidal categ ory ( A , p, j ) (previously called “premonoida l” in [1]) consists of a k -linear catego ry A and tw o k -linear functor s p : A op ⊗ A op ⊗ A / / V ect j : A / / V ect equipp e d with asso ciativity a nd unit constraints satisfying axioms (as describ ed in [1 ]) a nalogous to those used to define a monoida l structur e on A . The notion of a s ymmetric promonoida l catego ry (also int ro duced in [1]) was extended in [3] to that o f a braided promonoidal category . The main p oint is that (braided) promonoida l structures on A corres pond to co- contin uous (br a ided) mo noidal s tructures on the functor category [ A , V ect ]. This latter monoidal s tructure is often called the co n volution pro duct of A and V ect . Example 5.1 . Let ( A , p, j ) b e a small bra ided promo noidal catego ry with A ( I , I ) ∼ = I = k and j = A ( I , − ) , and supp ose that each hom-space A ( a, b ) is finite dimensional. Let f ∈ [ A , V ec t f ] be a very weak bialgebra in the conv o lution [ A , V ect ]. Supp ose a ls o that A ⊂ C where C is a sepa rable braide d monoidal category with p ( a, b , c ) ∼ = C ( a ⊗ b, c ) naturally; we suppo se the induced maps ( ‡ ) Z c p ( a, b , c ) ⊗ C ( c, C ) / / C ( a ⊗ b, C ) are isomor phisms (e.g., A mono idal). W e also supp ose that each a ∈ A has a dual a ∗ ∈ A . Then we have maps µ : f ∗ f / / f and η : k / / f I 12 BRIAN D A Y AND CRAIG P ASTRO and δ : f / / f ∗ f and ǫ : f I / / k satisfying asso ciativity a nd unital axioms. Define the functor U : C / / V ect by U ( C ) = Z a f a ⊗ C ( a, C ); then, by the Y oneda lemma, U ( a ∗ ) ∼ = U ( a ) ∗ if f ( a ∗ ) ∼ = f ( a ) ∗ for a ∈ A . Moreover, U is mo no idal and comonoidal on C via the maps r a nd i describ ed in the diagr am: U C ⊗ U D Z a,b f a ⊗ f b ⊗ C ( a, C ) ⊗ C ( b, D ) ∼ = / / Z a,b f a ⊗ f b ⊗ C ( a ⊗ b, C ⊗ D ) Z a,b f a ⊗ f b ⊗ Z c p ( a, b , c ) ⊗ C ( c, C ⊗ D ) Z c f c ⊗ C ( c, C ⊗ D ) , U ( C ⊗ D ) = o o r   i O O   C separable O O ( ‡ )   µ   δ O O Thu s, if f is separable, then so is U with dim U I = dim f I since U I = Z a f a ⊗ C ( a, I ) ∼ = f I by the Y oneda lemma, so tha t i 0 r 0 = dim U I · 1 if and only if ǫ η = dim f I · 1. Example 5.2 . Supp ose that ( A op , p, j ) is a small promonoidal ca tegory with I ∈ A s uc h that j ∼ = A ( − , I ) and with each x ∈ A an “a to m” in C (i.e., an ob ject x ∈ C for which C ( x, − ) pr e s erves all co limits) whe r e C is a co co mplete a nd co contin uous bra ided monoidal category co n ta ining A and each x ∈ A has a dual x ∗ ∈ A . Suppo se that the inclusio n A ⊂ C is dense ov er V ect ( that is, the canonical ev aluation morphism Z a C ( a, C ) · a / / C is a n isomo rphism for all C ∈ C ) and x ⊗ y ∼ = Z z p ( x, y , z ) · z ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 13 so that C ( a, x ⊗ y ) = C ( a, Z z p ( x, y , z ) · z ) ∼ = Z z p ( x, y , z ) ⊗ C ( a, z ) since a ∈ A is an atom in C , ∼ = p ( x, y , a ) by the Y oneda lemma applied to z ∈ A . Let W : A / / V ect b e a stro ng promonoidal functor on A . This mea ns that we have str ucture iso morphisms W x ⊗ W y ∼ = Z z C ( z , x ⊗ y ) ⊗ W z k ∼ = W I satisfying suitable asso ciativity a nd unital coher ence axio ms. Define the functor U : C / / V ect b y U C = Z a C ( a, C ) ⊗ W a. Then U ( x ∗ ) = Z a C ( a, x ∗ ) ⊗ W a ∼ = W ( x ∗ ) ∼ = W ( x ) ∗ , if W ( x ∗ ) ∼ = W ( x ) ∗ for all x ∈ A , and U I = Z a C ( a, I ) ⊗ W a ∼ = W I ∼ = k , so that i 0 r 0 = 1 a nd r 0 i 0 = 1. Also there ar e mutually inv ers e comp osite maps r and i given by: r : U C ⊗ U D ∼ = Z x,y C ( x, C ) ⊗ C ( y , D ) ⊗ U x ⊗ U y ∼ = Z x,y C ( x, C ) ⊗ C ( y , D ) ⊗ W x ⊗ W y ∼ = Z x,y C ( x, C ) ⊗ C ( y , D ) ⊗ Z z C ( z , x ⊗ y ) ⊗ W z ∼ = Z z C ( z , C ⊗ D ) ⊗ W z ∼ = U ( C ⊗ D ) , which uses the as sumptions that C is coco n tinuous mo noidal a nd A ⊂ C is dense. Thu s r i = 1 and ir = 1 so that U is a strong mono ida l functor. Example 5. 3. (See [5] Pro pos itio n 3.) Let C b e a braided compact monoidal category a nd le t A ⊂ C b e a full finite discrete Ca uc hy g enerator o f C which contains I and is clo sed under dualiza tion in C . As in the H¨ aring-O lden bur g case [5], we supp ose that each hom-space C ( C, D ) is finite dimens io nal with a chosen na tur al iso morphism C ( C ∗ , D ∗ ) ∼ = C ( C, D ) ∗ . 14 BRIAN D A Y AND CRAIG P ASTRO Then w e hav e a separable mono idal functor U C = M a,b ∈ A C ( a, C ⊗ b ) , whose structure ma ps ar e given by the comp osites r : U C ⊗ U D ∼ = M a,b,c,d C ( c, C ⊗ b ) ⊗ C ( a, D ⊗ d ) adjoint / / c = d o o M a,b,c C ( c, C ⊗ b ) ⊗ C ( a, D ⊗ c ) ∼ = M a,b C ( a, D ⊗ ( C ⊗ b )) ∼ = M a,b C ( a, ( D ⊗ C ) ⊗ b ) ∼ = M a,b C ( a, ( C ⊗ D ) ⊗ b ) = U ( C ⊗ D ) , and r 0 : k / / U I the diagonal, with i 0 its adjoint . Also U ( C ∗ ) = M a,b C ( a, C ∗ ⊗ b ) ∼ = M a,b C ( a ∗ , C ∗ ⊗ b ∗ ) ∼ = M a,b C ( a, C ⊗ b ) ∗ ∼ = U C ∗ for all C ∈ C . Example 5.4. Let ( A , p , j ) b e a finite bra ided promo no idal categ ory ov er Set f with I ∈ A such tha t j ∼ = A ( I , − ) and with a promo no idal functor d : A op / / Set f for whic h each structure map u : Z z p ( x, y , z ) × dz / / dx × dy is a n injection, and u 0 : dI / / 1 is a surjection. Then we have corresp onding maps Z z k [ p ( x, y , z )] ⊗ k [ dz ] / / / / o o o o k [ dx ] ⊗ k [ dy ] and k [ dI ] / / / / o o o o k [1] , where k [ s ] denotes the free k -vector space on the (finite) set s , in V e ct f . Define the functor U : C / / V ect f by U f = Z x f x ⊗ k [ dx ] ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 15 for f ∈ C = [ k ∗ A , V ect f ] (with the conv olution braided mono idal closed struc tur e) so that r : U f ⊗ U g =  Z x f x ⊗ k [ dx ]  ⊗  Z y g x ⊗ k [ dy ]  ∼ = Z x,y f x ⊗ g y ⊗ ( k [ dx ] ⊗ k [ dy ]) / / o o Z x,y f x ⊗ g y ⊗  Z z k [ p ( x, y , z )] ⊗ k [ dz ]  ∼ = Z z  Z x,y f x ⊗ g y ⊗ k [ p ( x, y , z )]  ⊗ k [ dz ] = Z z ( f ⊗ g )( z ) ⊗ k [ dz ] = Z z U ( f ⊗ g ) and i 0 : U I = Z x k [ A ( I , x )] ⊗ k [ dx ] ∼ = k [ dI ] / / o o k [1] ∼ = k . Hence i 0 r 0 = dim U I · 1 = | dI | · 1 . Thus, U b ecomes a separ able mono idal functor . Example 5.5. Let A b e a finite (discr ete) s e t and give the ca r tesian pro duct A × A the Set f -promonoida l str ucture corresp onding to bimo dule c o mpos ition (i.e., to matrix multiplication). If d : A × A / / Set f is a promo no idal functor, then its asso ciated structur e maps X z ,z ′ p (( x, x ′ ) , ( y , y ′ ) , ( z , z ′ )) × d ( z , z ′ ) = X z ,z ′ A ( z , x ) × A ( x ′ , y ) × A ( y ′ , z ′ ) × d ( z , z ′ ) ∼ = A ( x ′ , y ) × d ( x, y ′ ) / / d ( x, x ′ ) × d ( y , y ′ ) , and X z ,z ′ j ( z , z ′ ) × d ( z , z ′ ) = X z ,z ′ A ( z , z ′ ) × d ( z , z ′ ) ∼ = X z d ( z , z ) / / 1 , are deter mined by co mponents d ( x, y ′ ) / / / / d ( x, y ) × d ( y , y ′ ) d ( z , z ) / / / / 1 which g iv e A the structure o f a discrete co category over S et f . 16 BRIAN D A Y AND CRAIG P ASTRO Define the functor U : C = [ k ∗ ( A × A ) , V ect f ] / / V ect f by U f = M x,y ( f ( x, y ) ⊗ k [ d ( x, y )]) . Then we obtain mono idal and comonoida l structure maps U ( f ⊗ g ) i / / r o o U f ⊗ U g U I i 0 / / r 0 o o k ∼ = k [1] from the ca no nical maps M x,y ,z f ( x, z ) ⊗ g ( z , y ) ⊗ k [ d ( x, y )] z = u = v / / adjoint o o M x,u  f ( x, u ) ⊗ k [ d ( x, u )]  ⊗ M v, y  g ( v , y ) ⊗ k [ d ( v , y )]  and M z k [ d ( z , z )] / / o o k ∼ = k [1] . These giv e U the structure of a separable mono idal functor on C . 6. Concluding remark s If the origina l “fibre” functor U is faithful and exact then the T annak a equiv a- lence (dua lity) Lex( C op , V e ct ) ≃ Com o d (End ∨ U ) is av ailable. Th us, since C is braided mo noidal, so is Como d (End ∨ U ) with the tensor pro duct and unit induce d by the conv o lutio n pr o duct on Lex( C op , V e ct ); for conv enience we recall [2] that, for C co mpact, this con volution pro duct is g iv en by the res triction to Lex( C op , V e ct ) of the co end F ∗ G = Z C,D F C ⊗ GD ⊗ C ( − , C ⊗ D ) ∼ = Z C F C ⊗ G ( C ∗ ⊗ − ) computed in the whole functor category [ C op , V e ct ]. Moreov e r, when U is separable monoidal, the ca tegory Co (End ∨ U ) of cofree coactions of End ∨ U (as constructed in [6] for exa mple) also has a mo noidal structure ( Co (End ∨ U ) , ⊗ , k ), this time obtained from the algebr a structure of End ∨ U . The forg etful inclusion Como d (End ∨ U ) ⊂ Co (End ∨ U ) preserves colimits while Como d (End ∨ U ) has a small genera tor, namely { U C | C ∈ C } , and thus, from the sp ecial adjoint functor theorem, this inclusion has a r igh t adjoint. The v alue of the adjunction’s counit at the functor F ⊗ G in Co (End ∨ U ) is then a split monomorphism and, in particula r, the monoidal forgetful functor Como d (End ∨ U ) / / V ect , which is the comp osite Como d (End ∨ U ) ⊂ Co (End ∨ U ) / / V ect , is a separ able monoidal functor ex tens ion of the given functor U : C / / V ect . ON ENDOMORPHISM ALGEBRAS OF SEP ARABLE MONOIDAL FUNCTOR S 17 References [1] Br ian D a y . On closed categories of functors, in R ep orts of the Mi dwest Cate gory Seminar IV , Lecture Notes i n M athematics 137 (1970): 1–38. [2] Br ian J. Da y . Enriched T annak a reconstruct ion, J. Pure Appl. A lgebra 108 no. 1 (1996): 17–22. [3] B. Day , E. P anchadc haram, and R. Street. Lax braidings and the lax centre, i n Hopf Algebr as and Gener alizations , Cont emp orary M athemat i cs 441 (2007): 1–17. [4] Br ian Day and Ross Street. 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Dep ar tment of Ma thema tics, Macquarie Univ ersity, New South W ales 2109 Aus- tralia E-mail addr ess : craig@ics.mq .edu.au