Note on star-autonomous comonads

Note on star-autonomous comonads Craig P astro No v em b er 20 , 2018 Abstract W e develop an alternativ e approac h to star-autonomous comonads via linearly distributive categori es. It is shown that in th e autonomous case the notions of star-autonomous comonad and Hop f comonad coincide. 1 In tro duction Given a linearly distributiv e category C , this note determines wha t str ucture is required of a comonad G on C so that C G , the catego ry of Eilenberg- Moo r e coalgebr as of G , is aga in a linea rly distributive ca tegory . F urthermore, if C is equipp e d with negations (and is hence a star- autonomous c a tegory), the struc- ture requir ed to lift th e negations to C G is deter mined as well. This latter is equiv ale nt to lifting star-auto nom y and it is shown that the notion pr esen ted is equiv ale nt to a star-autonomous comona d [PS09]. As a cons equence of the pre - sentation given here, it may b e easily see n that a ny star- a utonomous co monad on an autonomo us ca teg ory is a Hopf monad [BV07]. 2 Lifting linear d istributivit y Suppo se C is a monoidal category and G : C → C is a comonad on C . Recall that C G , the catego ry of (Eilenberg -Moo re) co algebras o f G , is monoidal if and only if G is a monoida l comonad [M02]. In this section we are in terested in the structure required to lift linear distributivit y to the categor y of co algebras. A linearly distributive category C is a categor y eq uipped with tw o monoidal structures ( C , ⋆, I ) and ( C , ⋄ , J ), 1 and tw o compa tibilit y natura l transformations (called “linear distributions” ) ∂ l : A ⋆ ( B ⋄ C ) → ( A ⋆ B ) ⋄ C ∂ r : ( B ⋄ C ) ⋆ A → B ⋄ ( C ⋆ A ) , satisfying a larg e num b er of coherence diagr ams [CS97]. Suppo se G = ( G, δ, ǫ ) is a co monad on a linear ly distributiv e ca tegory C which is a mono ida l comonad on C with respect to b oth ⋆ a nd ⋄ , with structure 1 F or si mplicit y w e assume that the mo noidal structures are strict, although this is not necessary . F urthermore, in their original paper [ CS97] the tensor pr oducts ⋆ and ⋄ ar e r espec- tiv ely denoted b y ⊗ and ⋄ , and called t ensor and p ar , emphasizing their connec tion to linear logic. 1 maps ( G, φ, φ 0 ) and ( G, ψ , ψ 0 ) r espectively . If, for G -coalg ebras A , B , a nd C , the comonad G satisfies (1) GA ⋆ ( GB ⋄ GC ) GA ⋆ G ( B ⋄ C ) G ( A ⋆ ( B ⋄ C )) ( GA ⋆ GB ) ⋄ GC G ( A ⋆ B ) ⋄ GC G (( A ⋆ B ) ⋄ C ) , 1 ⋆ψ / / φ / / φ ⋄ 1 / / ψ / / ∂ l   ∂ l   it may b e seen that the morphism ∂ l bec omes a G -coa lgebra morphism. If G satisfies a similar axio m for ∂ r , i.e., (2) ( GB ⋄ GC ) ⋆ GA G ( B ⋄ C ) ⋆ GA G (( B ⋄ C ) ⋆ A ) GB ⋄ ( GC ⋆ GA ) GB ⋄ G ( C ⋆ A ) G ( B ⋄ ( C ⋆ A )) , ψ ⋆ 1 / / φ / / 1 ⋄ φ / / ψ / / ∂ r   ∂ r   then ∂ r also beco mes a G -coalgebr a morphism. Thus, Prop osition 2.1 . Given a line arly distributive c ate gory C and a c omonad G : C → C satisfying axioms (1) and (2) , the c ate gory C G is a line arly distributive c ate gory. Example 2.2. Let C b e a symmetric linearly distributive category and ( B , µ , η, δ, ǫ ) a bia lgebra in C with r espect to ⋄ . That is, the structure morphisms are g iven as µ : B ⋄ B → B δ : B → B ⋄ B η : J → B ǫ : B → J. Then, G = B ⋄ − is a comonad a nd is monoidal with resp ect to b oth ⋆ and ⋄ . The latter by I ∼ = J ⋄ I η ⋄ 1 − − → B ∗ I , and the following, ( B ⋄ U ) ⋆ ( B ⋄ V ) ∂ r − − − − − − → B ⋄ ( U ⋆ ( B ⋄ V )) 1 ⋄ (1 ⋆ c ) − − − − − − → B ⋄ ( U ⋆ ( V ⋄ B )) 1 ⋄ ∂ l − − − − − − → B ⋄ (( U ⋆ V ) ⋄ B ) 1 ⋄ c − − − − − − → B ⋄ ( B ⋄ ( U ⋆ V )) ∼ = − − − − − − → ( B ⋄ B ) ⋄ ( U ⋆ V ) µ ⋆ 1 − − − − − − → B ⋄ ( U ⋆ V ) . Rather large diagrams, which we leav e to the faith of the reader, prove that B ⋄ − satisfies (1) and (2), so that C B = Como d C ( B ), the categor y of como dules of B , is a linearly distributive category . 3 Lifting negations Suppo se no w that C is a linearly distributiv e catego ry equipp ed with negatio ns S and S ′ (corresp onding to ⊥ ( − ) and ( − ) ⊥ in [CS97]). That is, functor s S, S ′ : 2 C op → C together with the f ollowing (dinatural) ev aluation and coev aluation morphisms (3) S A ⋆ A e A − − → J A ⋆ S ′ A e ′ A − − → J I n A − − → A ⋄ S A I n ′ A − − → S ′ A ⋄ A, satisfying the four evident “triangle ident ities”. One such is  A ∼ = I ⋆ A n⋆ 1 − − → ( A ⋄ S A ) ⋆ A ∂ r − → A ⋄ ( S A ⋆ A ) 1 ⋄ e − − → A ⋄ J ∼ = A  = 1 A . If C is equipp ed with s uc h neg a tions we say simply that C is a line arly distributive c ate gory with ne gations . W e are interested to lift negations to C G . This means w e must ensure that the “negation” functors S, S ′ : C op → C lif t to functors ( C G ) op → C G , and the ev aluation and co ev aluation mor phisms ar e in C G , i.e., ar e G -coalgebr a morphisms. The following is es s en tially known from [S72]. Lemma 3.1. A ( c ontr avariant) functor S : C op → C may b e lifte d t o a funct or e S : ( C G ) op → C G such that the diagr am ( C G ) op C G C op C , e S / / U   U   S / / c ommu t es, if and only if ther e is a natur al tr ansformation ν : S → GS G satisfying the fol lowing two axioms (4) S GS G S G ν / / ǫ S G   S ǫ % % K K K K K K K K K K S GS G G 2 S G GS G G 2 S G 2 . ν / / δ S G / / ν % % K K K K K K K K K K Gν G / / G 2 S δ O O This may b e viewed as a distributive law of a contrav ariant functor ov er a comonad [S72]. In this c a se, we say that S m ay b e lifte d to C G , and a functor e S : ( C G ) op → C G is defined as e S ( A, γ ) =  S A, S A ν − → GS GA GS γ − − − → GA  e S ( f ) = S f . (T o see the r ev erse dir ection, s uppose ( A, γ ) is a coalgebra and e S is a functor C G → C G , so that e S A = ( S A, e γ ) is again a coalg e br a. Define ν := S A e γ − → GS A GS ǫ A − − − − → GS GA, which ma y b e seen to satisfy the axioms in (4).) W e will usually let the cont ext differentiate b etw een S and e S and simply write S in b oth cas e s . 3 Now, supp ose S and S ′ are equipp ed with natural trans fo rmations ν : S → GS G and ν ′ : S ′ → GS ′ G. such that they can b e lifted to C G . It remains to lift the e v aluation and co ev al- uation morphisms (3). Cons ide r the following ax io ms. (5) S A ⋆ GA S A ⋆ A J GS GA ⋆ G 2 A G ( S GA ⋆ GA ) GJ ν ⋆δ   φ / / Ge GA / / 1 ⋆ǫ / / e A / / ψ 0   (6) I GA ⋄ S GA GA ⋄ GS G 2 A G ( A ⋄ S G 2 A ) G ( A ⋄ S GA ) GI G ( A ⋄ S A ) n   1 ⋄ ν / / φ / / G (1 ⋄ S δ ) O O φ 0 / / Gn / / G (1 ⋄ S ǫ ) / / (7) GA ⋆ S ′ A A ⋆ S ′ A J G 2 A ⋆ GS ′ GA G ( GA ⋆ S ′ GA ) GJ δ⋆ν ′   φ / / Ge ′ GA / / ǫ⋆ 1 / / e ′ A / / ψ 0   (8) I S ′ GA ⋄ GA GS ′ G 2 A ⋄ GA G ( S ′ G 2 A ⋄ A ) G ( S ′ GA ⋄ A ) GI G ( S ′ A ⋄ A ) n ′   ν ′ ⋄ 1 / / φ / / G ( S ′ δ ⋄ 1) O O φ 0 / / Gn ′ / / G ( S ′ ǫ ⋄ 1) / / Prop osition 3.2. Supp ose C is a line arly distributive c ate gory with ne gation, G is a monoidal c omonad satisfying axioms (1) and (2) (so that C G is line arly distributive), and that S and S ′ may b e lifte d to C G . Then, G satisfies ax- ioms (5) , (6) , (7) , a nd (8) i f and o nly if C G is a line arly distributive c ate gory with n e gation. Pr o of. Supp ose ( A, γ ) is a G -coalgebra. W e start b y proving that axio m (5) holds if and only if e : S A ⋆ A → J is a G -coa lg ebra morphism. The following diagram prov es the “only if ” direction, S A ⋆ A GS GA ⋆ GA G ( S GA ⋆ A ) G ( S A ⋆ A ) S A ⋆ GA GS GA ⋆ G 2 A G ( S GA ⋆ GA ) GJ, S A ⋆ A J (5) 1   ν ⋆γ / / φ / / G ( S γ ⋆ 1) ' ' O O O O O O O O O 1 ⋆γ   1 ⋆Gγ   G (1 ⋆γ )   Ge   ν ⋆δ / / φ / / Ge ' ' O O O O O O O O O O O 1 ⋆ǫ   e / / ψ 0 / / 4 and this next diagr am the “ if ” direction S A ⋆ GA GS GA ⋆ G 2 A S A ⋆ A S GA ⋆ GA GS G 2 A ⋆ G 2 A GS GA ⋆ G 2 A G ( S GA ⋆ GA ) J GJ, 1 ⋆ǫ z z t t t t t t t t t S ǫ⋆ 1   ν ⋆δ / / GS Gǫ⋆ 1 { { w w w w w w w w 1 # # G G G G G G G G φ % % e $ $ J J J J J J J J J J e   ν ⋆δ / / GS δ ⋆ 1 / / φ / / Ge   ψ 0 / / where the b ottom square commutes a s e GA is a G -coalg ebra mor phism. Next we prov e that a xiom (6) holds if and o nly if n : I → A ⋄ S A is a G -coalgebr a morphism. The “only if ” direction is g iv en b y I A ⋄ S A GA ⋄ S GA GA ⋄ GS G 2 A G ( A ⋄ S G 2 A ) G ( A ⋄ S GA ) GI GA ⋄ S A GA ⋄ GS GA G ( A ⋄ S GA ) G ( A ⋄ S A ) , G ( A ⋄ S A ) (6) φ 0 / / Gn / / G (1 ⋄ S ǫ )   G (1 ⋄ S γ )   n $ $ J J J J J J J J J J 1 ⋄ ν / / φ / / G (1 ⋄ S δ ) / / n   γ ⋄ 1 $ $ J J J J J J J J J 1 ⋄ ν / / φ / / G (1 ⋄ S γ / / 1 ⋄ S γ   1 ⋄ GS Gγ   G (1 ⋄ S Gγ )   1   and the “if ” direction by I GA ⋄ S GA G 2 A ⋄ GS G 2 A G ( GA ⋄ S G 2 A ) G ( GA ⋄ S GA ) G ( A ⋄ S A ) GI GA ⋄ GS G 2 A G ( A ⋄ S G 2 A ) G ( A ⋄ S GA ) , φ 0 / / n   Gn   Gn & & M M M M M M M M M M δ ⋄ ν / / 1 ⋄ ν ( ( Q Q Q Q Q Q Q Q Q Q Q Q ψ / / Gǫ ⋄ 1   G (1 ⋄ S δ ) / / G ( ǫ ⋄ 1)   G ( ǫ ⋄ 1)   G (1 ⋄ Gǫ ) x x q q q q q q q q q ψ / / G (1 ⋄ S δ ) / / where the top squa re c o mm utes as n GA is a G -coalg ebra mor phism. The remaining tw o axioms are prov ed similarly . 4 Star-autonomous comonads Suppo se C = ( C , ⊗ , I ) is a star-auto nomous c a tegory . A sta r-autonomous comonad G : C → C is a comona d satisfying axioms (described be low) so that C G bec omes a s tar-autonomous categ ory [PS09]. In this section we show that co monads as in Prop osition 3.2 and star-a utonomous co monads coincide. W e r ecall the definition of star-autonomous comona d [PS09], but, as it suits our needs b etter here, we pres e n t a mor e symmetric version. First reca ll that a star-auto no mous catego ry may be defined as a monoidal category C = ( C , ⊗ , I ) equipp e d with an equiv a lence S ⊣ S ′ : C op → C such that (9) C ( A ⊗ B , S C ) ∼ = C ( A, S ( B ⊗ C )) , 5 natural in A, B , C ∈ C . The functor S is called the left star op er ation and S ′ the right star op er ation . By the Y oneda lemma, the isomorphism in (9) determines, a nd is deter mined by , the t wo following “ev aluation” morphisms: e = e A,B : S ( A ⊗ B ) ⊗ A → S B and e ′ = e ′ B ,A : B ⊗ S ′ ( A ⊗ B ) → S ′ A. Definition 4.1. A star-autonomous c omonad on a star-autono mous categor y C is a monoidal co mo nad G : C → C equipp ed with ν : S → GS G and ν ′ : S ′ → GS ′ G, satisfying (4) (i.e., S, S ′ may b e lifted to C G ), and this data must be such that the following four diagrams commu te. S S ′ G G GS GS ′ G GS S ′ ∼ = / / ∼ =   ν   GS ν ′ / / S ′ S G G GS ′ GS G GS ′ S ∼ = / / ∼ =   ν ′   GS ν / / S ( A ⊗ B ) ⊗ GA S ( A ⊗ B ) ⊗ A S B GS G ( A ⊗ B ) ⊗ G 2 A GS GB G ( S G ( A ⊗ B ) ⊗ GA ) G ( S ( GA ⊗ GB ) ⊗ GA ) 1 ⊗ ǫ / / e A,B / / ν   , , , , , , ν ⊗ δ        φ   , , , , , G ( S φ ⊗ 1) / / Ge GA,GB I I      GB ⊗ S ′ ( A ⊗ B ) B ⊗ S ′ ( A ⊗ B ) S ′ A G 2 B ⊗ GS ′ G ( A ⊗ B ) GS ′ GA G ( GB ⊗ S ′ G ( A ⊗ B )) G ( GB ⊗ S ′ ( GA ⊗ GB )) ǫ ⊗ 1 / / e ′ B,A / / ν ′   , , , , , δ ⊗ ν ′        φ   , , , , , G (1 ⊗ S ′ φ ) / / Ge ′ GB,GA I I      The first t wo diag r ams a bov e ensure that the equiv alence S ≃ S ′ lifts to C G , while the latter tw o diag rams ab ov e resp ectively e nsure that e and e ′ are G -coalgebr a morphisms, so that the isomorphism (9) also lifts to C G . W e wish to show that sta r-autonomous comona ds and co monads as in Pr opo- sition 3.2 coincide. It s hould not be surprising given the follo wing theorem. Theorem 4.2 ([CS97, Theorem 4.5]) . The notions of line arly distributive c at- e gories with ne gation and star-aut onomous c ate gories c oincide. Given a star-a utonomous c a tegory , iden tifying ⋆ := ⊗ (a nd the units I := I ⋆ = I ⊗ ) and defining (10) A ⋄ B := S ′ ( S B ⋆ S A ) ∼ = S ( S ′ B ⋆ S ′ A ) J := S I ∼ = S ′ I 6 gives a linea rly distributive c a tegory [CS97]. The neg ations of cours e come from S and S ′ . In [CS97], they co nsider the symmetric case, but the co r resp ondence betw een linearly distributiv e categor ies with neg a tion and star-autonomo us cat- egories holds in the nonco mm utative ca se a s w ell. Now, given Theorem 4.2, Pro positio n 3.2 says that if C is s tar-autonomous, and G is such a comonad, then C G is star-auto no mous. W e now compar e the t wo definitions. Suppo se now that G is a como nad on a linea r distr ibutiv e ca tegory C as in Prop osition 3.2. W e wish to show that it is a star-autono mous comonad. Rather than proving the axioms, it is simpler to s ho w directly that the morphisms under consideratio n are G -coalgebra morphisms. T o this end, the equiv alence S ≃ S ′ is given by the equations A ∼ = I ⋆ A n ′ S A ⋆ 1 − − − − → ( S ′ S A ⋄ S A ) ⋆ A ∂ r − → S ′ S A ⋄ ( S A ⋆ A ) 1 ⋄ n − − → S ′ S A ⋄ J ∼ = S ′ S A and S ′ S A ∼ = I ⋆ S ′ S A n A ⋆ 1 − − − → ( A ⋄ S A ) ⋆ S ′ S A ∂ r − → A ⋄ ( S A ⋆ S ′ S A ) 1 ⋄ e ′ S A − − − − → A ⋄ J ∼ = A, and e A,B and e ′ B ,A are resp ectively defined a s S ( A ⋆ B ) ⋆ A S ( A ⋆ B ) ⋆ A ⋆ I S ( A ⋆ B ) ⋆ A ⋆ ( B ⋄ S B ) ( S ( A ⋆ B ) ⋆ A ⋆ B ) ⋄ S B J ⋄ S B S B ∼ =   1 ⋆ 1 ⋆n   ∂ l / / e A⋆B ⋄ 1 O O ∼ = O O e A,B / / B ⋆ S ′ ( A ⋆ B ) I ⋆ B ⋆ S ′ ( A ⋆ B ) ( S ′ A ⋄ A ) ⋆ B ⋆ S ′ ( A ⋆ B ) S ′ A ⋄ ( A ⋆ B ⋆ S ′ ( A ⋆ B )) S ′ A ⋄ J S B ∼ =   n ′ ⋆ 1 ⋆ 1   ∂ r / / 1 ⋄ e ′ A⋆B O O ∼ = O O e ′ B,A / / In the s ituation of Pro positio n 3.2, we see that all four o f these morphisms are given as co mposites of G - coalgebra mo rphisms, a nd thus, ar e G -coalg ebra morphisms themselves. Therefore, G is a star-auto nomous como nad. In the other direction s upp ose G is a star-autono mous comonad o n a star- autonomous category C . It is similar to show that it is a comona d satisfying the requir emen ts of Prop osition 3.2. Using the identifications in (1 0), the t wo linear distributions are defined a s follo ws. A ⋆ ( B ⋄ C ) A ⊗ S ′ ( S C ⊗ S B ) A ⊗ S ′ ( S C ⊗ S ( A ⊗ B ) ⊗ A ) S ′ ( S C ⊗ S ( A ⊗ B )) ( A ⋆ B ) ⋄ C ∂ l / / ∼ =   1 ⊗ S ′ (1 ⊗ e ) # # F F F F F F F e ′ ; ; x x x x x x x ∼ = O O ( B ⋄ C ) ⋆ A S ( S ′ C ⊗ S ′ B ) ⊗ A S ( A ⊗ S ′ ( C ⊗ A ) ⊗ S ′ B ) ⊗ A ) S ( S ′ ( C ⊗ A ) ⊗ S ′ B ) B ⋄ ( C ⋆ A ) ∂ r / / ∼ =   S ( e ′ ⊗ 1) ⊗ 1 # # F F F F F F F e ; ; x x x x x x x ∼ = O O 7 The ev aluation ma ps e A and e ′ A are defined as e A,I and e ′ A,I , and the co ev alu- ation maps n A and n ′ A as n A =  I ∼ = S S ′ I S e ′ A,I − − − − → S ( A ⊗ S ′ A ) = A ⋄ S A  n ′ A =  I ∼ = S ′ S I S ′ e A,I − − − − → S ′ ( S A ⊗ A ) = S ′ A ⋄ A  Again, each mo rphism is a G -coalg ebra morphism, or comp osite thereof, and therefore is itself a G -coa lgebra morphism. Thu s, b oth notions coincide, and w e will simply call either notio n a star- autonomous c omonad , and let context differ e n tiate the axio matization. Example 4. 3 . Any Hopf alg ebra H in a star-autono mous catego ry C gives rise to a star -autonomous como na d H ⊗ − : C → C . See [P S09, pg. 3515] for details. Example 4.4. If C is a symmetric clos ed monoidal catego ry with finite pro d- ucts, then we ma y apply the C hu construction [B7 9] to produce a star-autono mous category Chu( C ). C fully faithfully embeds int o Chu( C ), C ֒ → Ch u( C ) and this functor is strong symmetric monoidal. Thus, a n y Hopf algebra in C bec omes a Hopf algebra in Chu( C ), and th us, an exa mple of a sta r-autonomous comonad. 5 The compact case ⋆ = ⋄ If C is a linear ly distr ibutiv e categor y with negation for whic h ⋆ = ⋄ (and thus, I = J ), then C is an autonomo us (= r ig id) category . The functor S provides left duals, while S ′ provides right dua ls. It is no t hard to see that in this case, any star -autonomous mona d G (after dualiz ing ) is a Hopf monad [BV07]. Set ⋆ = ⋄ and I = J and dualize a xioms (5), (6), (7 ), and (8). They co rresp ond in [B V07] to axioms (23), (22), (21 ), and (20) resp ectively . (In their no tation ∨ ( − ) = S a nd ( − ) ∨ = S ′ .) There fo re, we hav e: Prop osition 5.1. Star-aut onomous m onads on autonomous c ate gories ar e Hopf monads. Ac kno wledgeme n ts I w ould like to thank Robin Co c kett and Masahito Hasegawa for their v a luable suggestions . References [B79] Michael Barr. ∗ -Autonomous categor ies, V o lume 752 of Lec tur e Notes in Mathematics. Springer, Berlin, 1979 . With an app endix by Po Hsiang Ch u. [BV07] Alain Brugui` eres and Alexis Vir e lizier. Hopf monads, Adv a nces in Ma th- ematics 215 no. 2 (200 7) 67 9–733. 8 [CS97] J.R.B. Co ck ett and R.A.G. Seely . W eakly distributive categories, Jour - nal of Pure and Applied Algebra 11 4 (1997) 13 3–173. Corrected version av a ilable fro m the second authors webpage. [M02] I. Mo erdijk. Monads on tens o r catego ries, J ournal of P ure and Applied Algebra 168 (2002 ) 18 9–208. [PS09] Craig Pastro and Ross Street. Clos ed ca tegories, sta r-autonomy , and monoidal comonads , Jo ur nal o f Algebra 321 no. 11 (2009 ) 349 4–3520 . [S72] Ross Street. The formal theory of monads, Journa l of Pure a nd Applied Algebra 2 no. 2 (1972 ) 149–168. Department of Mathematics, Kyushu Univers ity , 744 Motook a, N ishi-ku, F ukuoka 819- 0395, Japan craig@math .kyushu-u.ac.j p 9