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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 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.

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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

IM⊗m m⊗IM

M ⊗ M

m

• M ⊗ M

m

M

I ⊗ M

lM

• u⊗IM
• M

M ⊗ I

rM

• IM⊗u
• 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(M0)= the category of comonoids in M, or

monoids in the opposite category.

(Classical Examples)

• Mon(Set): usual monoids in Set;
• Mon(VectK) =K-Algebras;

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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 U0 : Mon(M0) → M0 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.

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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 = VectK, R. Block, P. Leroux, J. Pure Appl. Algebra ’85.

(construction)

• M = VectK T. Fox, J. Pure Appl. Algebra ’93. (different construction)
• M = CrgA = CoMon(AMA) M. Hazewinkel J. Pure Appl. Algebra ’03;

Cofree corings exist over V = An;

• M = CrgA A. Agore, Proceedings of the AMS, ’11. Open question: “Is

there a cofree A-coring over any A-bimodule?”

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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.

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Investigating the (SAFT)

Proposition

Let M be a monoidal category, CoMon(M) be the category of comonoids

• f 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).

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Existence of Cofree Corings

Theorem

(i)CrgA (= CoMon(AMA) ) is generated by all corings of cardinality ≤ max{|A|, ℵ0}. (ii) U : CrgA → AMA has a right adjoint. Hence, there is a cofree coring C(V ) on every A-bimodule V . C(V ) = lim

− →

f :U(G)→V | G∈CrgA; |G|≤{|A|,ℵ0}

G

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CoAlg(HM) and CoAlg(MH)

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(HM) and CoAlg(MH) are cocomplete, co-wellpowered, and the forgetful functors F H : CoAlg(MH) − → MH and FH : CoAlg(HM) − → HM preserve colimits.

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Existence of Cofree Coalgebras in CoAlg(HM)

Proposition

The left H-module coalgebras f .g.CoAlg(HM) which are finitely generated as left H-modules form a system of generators for CoAlg(HM). Consequently, the functor FH : CoAlg(HM) → HM has a right adjoint. GH(V ) = lim

− →

[f :D→V ]∈HM, D∈f .g.CoAlg(HM)

D.

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Existence of Cofree Coalgebras in CoAlg(MH)

Theorem

The category CoAlg(MH) (=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

− →

[f :D→V ]∈MH, D∈fin.dim.CoAlg(MH)

D.

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Explicit Description for Generators in CoAlg(MH)

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(MH) of finite dimensional algebras in MH (and also in Alg(MH)). The coalgebras V ∗ ⊗ V form a system of generators of CoAlg(MH) (= the category of H-comodule algebras).

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Thank You

Thank You!

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