GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS

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1 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

2 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 constraint): 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

3 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: M M M I M m M M m I M m M M m M I M l M M u I M m M M r M I M u M I Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 3 / 14

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 Thank You Thank You! Adnan H Abdulwahid GENERATORS FOR COMONOIDS AND UNIVERSAL CONSTRUCTIONS 14 / 14