GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS Adnan H Abdulwahid University of Iowa Third Conference on Geometric Methods in Representation Theory University of Iowa Department of Mathematics November 24, 2014 Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 1 / 14
Monoidal Categories A monoidal category is a tuple ( M , ⊗ , I , a , l , r ), where M is a category ⊗ : M × M → M is a bifunctor called ( tensor product ) I is an object in M called ( unit ) of M a is a functorial isomorphism called ( associativity constraint ): a X , Y , Z : ( X ⊗ Y ) ⊗ Z → X ⊗ ( Y ⊗ Z ) l is a functorial isomorphism called ( left unit constrain t): l X : I ⊗ X → X r is a functorial isomorphism called ( right unit constraint ): r X : X ⊗ I → X The functorial morphisms a , l , and r satisfy the coherence axioms. Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 2 / 14
� � � � � � Monoids Let ( M , ⊗ , I , a , l , r ) be a monoidal category. A monoid is a triple ( M , m , u ), where M is an object in M , and m : M ⊗ M → M (multiplication) u : I → M (unit) are morphisms in M subject to the associativity and unity axioms: I M ⊗ m � l M r M M ⊗ M ⊗ M M ⊗ M I ⊗ M M M ⊗ I m ⊗ I M � m m u ⊗ I M I M ⊗ u m � M M ⊗ M M ⊗ M Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 3 / 14
Notations and Examples for Monoids • Mon ( M )=the category of monoids in M . • CoMon ( M ) := Mon ( M 0 )= the category of comonoids in M , or monoids in the opposite category. (Classical Examples) - Mon ( Set ): usual monoids in Set ; - Mon ( Vect K ) = K -Algebras; Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 4 / 14
Basic Questions Question Let ( M , ⊗ , I , a , l , r ) be a monoidal category. (1) When does U : Mon ( M ) → M have a left adjoint? (2) When does U 0 : Mon ( M 0 ) → M 0 have a left adjoint? Equivalently, When does U : CoMon ( M ) → M have a right adjoint? • The free monoid and Mac Lane’s Observation. • Cofree and the dual of Mac Lane’s Observation. Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 5 / 14
A little history Question Given a monoidal category M , when does U : CoMon ( M ) → M have a right adjoint? • M = R − Mod ,the category of modules over commutative ring, M. Barr, J. Algebra ’74. (existence) • M = Vect K , R. Block, P. Leroux, J. Pure Appl. Algebra ’85. (construction) • M = Vect K T. Fox, J. Pure Appl. Algebra ’93. (different construction) • M = Crg A = CoMon ( A M A ) M. Hazewinkel J. Pure Appl. Algebra ’03; Cofree corings exist over V = A n ; • M = Crg A A. Agore, Proceedings of the AMS, ’11. Open question: “Is there a cofree A -coring over any A -bimodule?” Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 6 / 14
The Special Adjoint Functor Theorem (SAFT) (The Dual Version) Theorem (SAFT) If A is a cocomplete, co-wellpowered category and with a generating set, then every cocontinuous functor from A to a locally small category has a right adjoint. Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 7 / 14
Investigating the (SAFT) Proposition Let M be a monoidal category, CoMon ( M ) be the category of comonoids of M and U : CoMon ( M ) → M be the forgetful functor. (i) If M is cocomplete, then CoMon ( M ) is cocomplete and U preserves colimits. (ii) If furthermore M is co-wellpowered, then so is CoMon ( M ) . Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 8 / 14
Existence of Cofree Corings Theorem (i)Crg A ( = CoMon ( A M A ) ) is generated by all corings of cardinality ≤ max {| A | , ℵ 0 } . (ii) U : Crg A → A M A has a right adjoint. Hence, there is a cofree coring C ( V ) on every A-bimodule V . C ( V ) = lim G − → f : U ( G ) → V | G ∈ Crg A ; | G |≤{| A | , ℵ 0 } Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 9 / 14
CoAlg ( H M ) and CoAlg ( M H ) We note that if M is an abelian monoidal category, then CoAlg ( M ) = CoMon ( M ). Proposition Let H be a bialgebra over a field K . The categories of coalgebras CoAlg ( H M ) and CoAlg ( M H ) are cocomplete, co-wellpowered, and the forgetful functors F H : CoAlg ( M H ) − → M H and F H : CoAlg ( H M ) − → H M preserve colimits. Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 10 / 14
Existence of Cofree Coalgebras in CoAlg ( H M ) Proposition The left H-module coalgebras f . g . CoAlg ( H M ) which are finitely generated as left H-modules form a system of generators for CoAlg ( H M ) . Consequently, the functor F H : CoAlg ( H M ) → H M has a right adjoint. G H ( V ) = lim D . − → [ f : D → V ] ∈ H M , D ∈ f . g . CoAlg ( H M ) Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 11 / 14
Existence of Cofree Coalgebras in CoAlg ( M H ) Theorem The category CoAlg ( M H ) (=right H-comodule coalgebras) is generated by objects which are finite dimensional. Consequently, F H has a right adjoint G H given by G H ( V ) = lim D . − → [ f : D → V ] ∈M H , D ∈ fin . dim . CoAlg ( M H ) Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 12 / 14
Explicit Description for Generators in CoAlg ( M H ) Theorem Let H be a Hopf algebra over a field K . The finite dimensional algebras of the form V ⊗ V ∗ for finite dimensional H-comodules V , form a system of cogenerators in the category fdAlg ( M H ) of finite dimensional algebras in M H (and also in Alg ( M H ) ). The coalgebras V ∗ ⊗ V form a system of generators of CoAlg ( M H ) (= the category of H-comodule algebras). Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 13 / 14
Thank You Thank You! Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 14 / 14
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