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Frobenius algebras and monoidal categories Ross Street Annual Meeting Aust. Math. Soc. September 2004 The Plan Step 1 Recall the ordinary notion of Frobenius algebra over a field k . Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e : A æ æÆ k such that: e(ab) = 0 for all a Œ A implies b = 0. Example A = M n (k ) = the algebra of n ¥ n matrices over k e(a) = the trace Tr(a) of a . More generally, for any Frobenius algebra A , we can enrich the algebra M n ( A) with the Frobenius e Tr structure M n ( A) ææÆ A æ æÆ k . It follows, using Wedderburn dimensional Theory, that semisimple Frobenius structure. 2 every finite- algebra admits a Example X an n-dimensional oriented compact manifold m H ( X) = de Rham cohomology of X of degree m = closed differentiable m-forms on X modulo exact forms. n * H (X) = ≈ H m (X) m=0 is a real algebra under wedge product integration ÚX : H* ( X ) æ æÆ R over X provides a Frobenius structure. 3 Monoidal categories A category V is m o n o i d a l when it is equipped æÆV (called the tensor with a functor ƒ : V ¥ V æ of V (called the tensor product), an object I unit), and three natural families of isomorphisms ( A ƒ B) ƒ C @ A ƒ ( B ƒ C) , I ƒ A @ A @ A ƒ I in V (called associativity and unital constraints), such that the pentagon, involving the five ways of bracketing four objects, commutes, and the associativity constraint with B = I is compatible with the unit constraints. Example V = Vect k = the category of k-linear spaces with usual tensor product G Example V = Vect k = Re pk G = the category of k- linear representations of the group G with usual tensor product 4 Braided and symmetric monoidal categories Call V braided when it is equipped with a natural family of isomorphisms cA,B : A ƒ B @ B ƒ A (called the braiding) satisfying two conditions (one expressing c AƒB,C in terms of associativity constraints, 1A ƒ c B , C and c C , A ƒ 1B , and a similar one for c A , Bƒ C ). A braiding is a symmetry when c B , A o c A , B = 1A ƒ B . Example Vect k is symmetric as is the more general Re p k G 5 String diagrams Morphisms f : A ƒ B ƒ C æ æÆ D ƒ E in a monoidal category V can be represented by diagrams in the Euclidean plane: B A C f D E . The strings are labelled by objects and the nodes are labelled by morphisms. Composition of morphisms is performed vertically while tensoring is horizontal, creating more complicated plane graphs. This geometric calculus in the plane faithfully represents calculations in monoidal categories. We shall see how this works as we continue. 6 Monoids in a monoidal category A m o n o i d in V is an object A equipped with a "multiplication" m : A ƒ A æÆ æ A and a "unit" h : I æÆ æ A satisfying the associativity condition: m m = m m and the unit condition: h h m = = m Here all strings are labelled by A . 7 Examples ∑ A monoid in the category Set of sets, where the tensor product is cartesian product, is a monoid in the usual sense. ∑ If we use the coproduct (disjoint union) in Set as tensor product, every set has a unique monoid structure. ∑ A monoid in Vect k , with the usual tensor product of vector spaces, is precisely a k-algebra; monoids in monoidal k-linear categories are also sometimes called algebras. ∑ op A monoid in the dual category Vect k , with the usual tensor product of vector spaces, is precisely a k-coalgebra. ∑ A monoid in the category Cat of categories (where the morphisms are functors and the tensor product is cartesian product) is a strict monoidal category. 8 Duality within a monoidal category A duality A JB between two objects A and B i n a monoidal category V is a pair of morphisms a:AƒBæ æÆ I and b : I æ æÆ B ƒ A called the counit and unit, respectively, such that A b a A = B A and A B b B A a = B monoidal category is called a u t o n o m o u s (compact or rigid) when for every object A there exist B and C with C Example We have A JB JA JB. in Vect k for some B if and only if A is finite dimensional; in this case, A* where JA J A A* = Vect k ( A , k ) functions from A to k. 9 * is the space of linear Frobenius monoids in a monoidal category Theorem Suppose A is a monoid in V a n d e:Aæ æÆ I is a morphism. The following six conditions are equivalent and define Frobenius monoid: (a) there exists r : I æ æÆ A ƒ A such that (A ƒ m) o (r ƒ A) = (m ƒ A) o (A ƒ r) and ( A ƒ e ) o r = h = (e ƒ A) o r ; æÆ A ƒ A such that (b) there exists d : A æ ( A ƒ m ) o (d ƒ A ) = d o m = ( m ƒ A ) o ( A ƒ d ) and (A ƒ e) o d = 1A = (e ƒ A) o d ; (c) there exists d : A æ æÆ A ƒ A such that ( A, e , d ) is a comonoid and ( A ƒ m ) o (d ƒ A ) = d o m = ( m ƒ A ) o ( A ƒ d ) ; æÆ I exists for a duality (d) a counit s : A ƒ A æ A A w i t h s o ( A ƒ m ) = s o (m ƒ A ) ; J (e) s = e o m is a counit for A JA; (f) the free functor F : V æ æÆV A is right adjoint to the forgetful functor Example If B U :V A æ æÆV with counit e . JAJB Frobenius monoid in V. 10 then AƒB is a The self-dual nature of Frobenius monoid Part (c) of the Theorem: A Frobenius algebra consists of a monoid and comonoid structure on A subject to the condition d m m = = d d m Invertibility of Frobenius monoid morphisms If f : A æ æÆ B is both a monoid and comonoid morphism then it has inverse represented by B h m f e d A 11 Commutative Frobenius monoids Assume V is braided. A monoid A in V is c o m m u t a t i v e when c A,A A ƒ A ææ æÆ A ƒ A m m A . A comonoid A in V is c o c o m m u t a t i v e when A d d c A,A A ƒ A ææ æÆ A ƒ A . Proposition For a Frobenius monoid, equivalent to cocommutativity. 12 commutativity is The group algebra G finite group m A A = kG d A ƒ A ææ1 Æ A A ææ1 Æ A ƒ A g ƒ h a gh g a gƒg d 2 A ææ ÆA ƒ A cocom. Hopf m 2 A ƒ A ææ ÆA com. A Ï g for g = h Hopf -1 g a Âgh ƒ h g ƒ h a Ì Ó 0 otherwise h * Frobenius commutative and cocommutative Frobenius Moreover, the lower right square is a commutative and cocommutative Frobenius algebra in Re p k G . Larson-Sweedler: Every finite-dimensional Hopf algebra admits a Frobenius structure. However: the coalgebra structure of the Frobenius structure is not that of the Hopf algebra. 13 2D Topological Quantum Field Theories There is a symmetric monoidal category 2Cob of 2-cobordisms: objects are natural numbers; a morphism M : n æ æÆ m is an oriented twodimensional cobordism whose boundary consists of n circles with inward orientation and m circles with outward orientation, where two morphisms are identified when there is an orientationpreserving diffeomorphism between them. 4 3 composition is vertical stacking when target of one and source of other match; tensoring is horizontal placement. A 2D topological quantum field theory is a symmetric strong monoidal functor T : 2Cob æ æÆ Vect k . 14 A universal commutative Frobenius monoid Theorem In 2Cob there is a commutative Frobenius algebra 1 object multiplication unit comultiplication counit . Every commutative Frobenius m o n o i d A in any symmetric monoidal category V is the v a l u e A = T1 of an essentially unique symmetric strong æÆV . monoidal functor T : 2Cob æ Indeed, evaluation at 1 determines an equivalence of groupoids ~ CommFrob(V ). SymmStMon(2Cob,V ) – Corollary 2D topological quantum field theories are determined up to isomorphism by c o m m u t a t i v e Frobenius algebras. 15 Modules There is a monoidal bicategory Vect k - Mod : objects are k-linear categories A , B , . . . ; morphisms M A ææ Æ B are k-linear functors M : B op ƒ A æ æÆ Vect k (called modules from A t o B ) ; 2-cells are natural transformations; M N composition of modules A ææÆ B ææÆ C has ( N o M)(C, A) defined as the coequalizer of ’ B,B¢ M(B¢ , A) ƒ B (B, B¢) ƒ N(C, B) æ æÆ M(B, A) ƒ N(C, B) æ æÆ B , B ¢ ’ (called tensor product over B ) ; tensor product A ƒ B is defined by ob(A ƒ B ) = obA ¥ obB (A ƒ B )((A , B),(A¢ , B¢) = A (A , A¢) ƒ B (B, B¢) . A op behaves like a dual for vector spaces: there is an equivalence between modules A ƒB æ æÆ C and modules B æ æÆ A op ƒ C . 16 Frobenius monoidal categories Just as we looked at monoids in monoidal categories, we look at pseudomonoids in monoidal bicategories. In Vect k - Mod the pseudomonoids include monoidal k-linear categories such as Vect k itself. The Frobenius requirement is related to the notion of star-autonomy due to Michael Barr. Every rigid (autonomous, compact) monoidal category is star-autonomous. In particular, Vect k is Frobenius. Quantum groupoids provide further examples of Frobenius pseudomonoids. For further details: [DS2] Brian Day and Ross Street, Quantum categories, star autonomy, and quantum groupoids, F i e l d s Institute Communications 43 (Amer. Math. Soc. 2004) 193-231. [St] Ross Street, Frobenius monads and pseudomonoids, J. Math. Physics 45(10)(to appear October 2004). 17 Bibliography [Ba1] M. Barr, * -Autonomous categories, with an appendix by Po Hsiang Chu. Lecture Notes in Mathematics 752 (Springer, Berlin, 1979). [Ba2] M. Barr, Nonsymmetric * -autonomous categories, Theoretical Computer Science 139 (1995) 115-130. [Bu] J. 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[Nes] Cecil Nesbitt, On the regular representations of algebras, Ann. Math. 39 (1938) 634–658. [St] Ross Street, Frobenius monads and pseudomonoids, J. Math. Physics 45(10)(to appear October 2004). Centre of Australian Category Theory Macquarie University email: [email protected] 19