The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We …
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited …
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free omega categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a composite of atomic cells, this is the sense in which the omega category is free. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections [Street1994] were required some years following the original publication. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.
The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it …
The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the `many-object' setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.
The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are …
The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are given a specific orientation and satisfy Mac Lane's five axioms. Whilst recent applications justify the use of skew-monoidal structure, they do not give an intrinsic justification for the form the structure takes (the orientation of the constraints and the axioms that they satisfy). This paper provides a perspective on skew-monoidal structure which, amongst other things, makes it quite apparent why this particular choice is a natural one. To do this, we use the Catalan simplicial set C. It turns out to be quite easy to describe: it is the nerve of the monoidal poset (2, v, 0) and has a Catalan number of simplices at each dimension (hence the name). Our perspective is that C classifies skew-monoidal structures in the sense that simplicial maps from C into a suitably-defined nerve of Cat are precisely skew-monoidal categories. More generally, skew monoidales in a monoidal bicategory K are classified by maps from C into the simplicial nerve of K.
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We …
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [B\oh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.
We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular omega-category.
We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular omega-category.
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited …
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free $\omega$-categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a (unique) composite of atomic cells. This is the sense in which the $\omega$-category is (freely) generated from its atoms. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections were required some years following the original publication~\cite{Street1994}. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond …
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited …
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free $\omega$-categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a composite of atomic cells, this is the sense in which the $\omega$-category is free. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections [Street1994] were required some years following the original publication. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.
The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it …
The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the `many-object' setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We …
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [B\oh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited …
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free omega categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a composite of atomic cells, this is the sense in which the omega category is free. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections [Street1994] were required some years following the original publication. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.
We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular omega-category.
We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular omega-category.
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited …
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free $\omega$-categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a (unique) composite of atomic cells. This is the sense in which the $\omega$-category is (freely) generated from its atoms. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections were required some years following the original publication~\cite{Street1994}. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited …
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free $\omega$-categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a composite of atomic cells, this is the sense in which the $\omega$-category is free. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections [Street1994] were required some years following the original publication. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that require the most complicated proofs, we are able to investigate where this theory might benefit from further study and which results need to be considered most carefully in future work.
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We …
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.
The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are …
The basic data for a skew-monoidal category are the same as for a monoidal category, except that the constraint morphisms are no longer required to be invertible. The constraints are given a specific orientation and satisfy Mac Lane's five axioms. Whilst recent applications justify the use of skew-monoidal structure, they do not give an intrinsic justification for the form the structure takes (the orientation of the constraints and the axioms that they satisfy). This paper provides a perspective on skew-monoidal structure which, amongst other things, makes it quite apparent why this particular choice is a natural one. To do this, we use the Catalan simplicial set C. It turns out to be quite easy to describe: it is the nerve of the monoidal poset (2, v, 0) and has a Catalan number of simplices at each dimension (hence the name). Our perspective is that C classifies skew-monoidal structures in the sense that simplicial maps from C into a suitably-defined nerve of Cat are precisely skew-monoidal categories. More generally, skew monoidales in a monoidal bicategory K are classified by maps from C into the simplicial nerve of K.
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond …
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes.We make these constructions into a functor on a category of chain complexes with …
There are several ways to construct omega-categories from combinatorial objects such as pasting schemes or parity complexes.We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes.This functor from augmented directed complexes to omega-categories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases.The omega-categories equivalent to augmented directed complexes with good bases include the omega-categories associated to globes, simplexes and cubes; thus the morphisms between these omega-categories are determined by morphisms between chain complexes.It follows that the entire theory of omega-categories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omega-categories and calculate some internal homomorphism objects.
Kornel Szlachanyi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized …
Kornel Szlachanyi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are skew-monoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V ) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures.
We associate, in a functorial way, a monoidal bicategory $\mathsf{Span}| \mathcal V$ to any monoidal bicategory $\mathcal V$. Two examples of this construction are of particular interest: Hopf polyads (due …
We associate, in a functorial way, a monoidal bicategory $\mathsf{Span}| \mathcal V$ to any monoidal bicategory $\mathcal V$. Two examples of this construction are of particular interest: Hopf polyads (due to Brugui\`eres) can be seen as Hopf monads in $\mathsf{Span}| \mathsf{Cat}$ while Hopf group monoids in a braided monoidal category $V$ (in the spirit of Turaev and Zunino), and Hopf categories over $V$ (by Batista, Caenepeel and Vercruysse) both turn out to be Hopf monads in $\mathsf{Span}| V$. Hopf group monoids and Hopf categories are Hopf monads on a distinguished type of monoidales fitting the framework studied recently by B\ohm and Lack. These examples are related by a monoidal pseudofunctor $V\to \mathsf{Cat}$.
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We …
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.
This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets …
This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets of C*-algebras in Minkowski space for applications to relativistic quantum field theory: Roberts' additional challenge was that `no amount of staring at the low dimensional cocycle conditions would reveal the pattern for higher dimensions'. This report takes up this challenge, presenting a natural inductive construction of explicit cubical cocyle conditions, and gives three ways in which the simplicial ones can be derived from these. (A dual string-diagram version of this work, giving rise to a Pascal's triangle of diagrams for cocycle conditions, has been described elsewhere by Street). A consequence of this work is that the Yang-Baxter equation, the `pentagon of pentagons', and higher simplex equations, are in essence different manifestations of the same underlying abstract structure. There has been recent interest in higher-categories, by computer scientists investigating concurrency theory, as well as by physicists, among others. The dual `string' version of this paper makes clear the relationship with higher-dimensional simplex equations in physics. Much work in this area has been done since these notes were written: no attempt has been made to update the original report. However, all diagrams have been redrawn by computer, replacing all original hand-drawn pictures.
Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous pseudoalgebras …
Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only if it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What it means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to “strongly separable” Frobenius algebras and “weak monoidal Morita equivalence.” Wreath products of Frobenius algebras are discussed.
Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related. (0.1) Noncontradiction : Namely, to provide a formal frame rich …
Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related. (0.1) Noncontradiction : Namely, to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent, or at least relatively consistent with a well-established and “safe” theory, e.g. Zermelo-Frankel (ZF). (0.2) Adequacy , in the following, nontechnical sense: (i) The basic notions must be simple enough to make transparent the syntactic structures involved. (ii) The translation between the formal language and the usual language must be, or very quickly become, obvious. This implies in particular that the terminology and notations in the formal system should be identical, or very similar, to the current ones. Although this may seem minor, it is in fact very important. (iii) “Foundations” can only be “foundations of a given domain at a given moment”, therefore the frame should be easily adaptable to extensions or generalizations of the domain, and, even better, in view of (i), it should suggest how to find meaningful generalizations. (iv) Sometimes (ii) and (iii) can be incompatible because the current notations are not adapted to a more general situation. A compromise is then necessary. Usually when the tradition is very strong (ii) is predominant, but this causes some incoherence for the notations in the more general case (e.g. the notation f ( x ) for the value of a function f at x obliges one, in category theory, to denote the composition of arrows ( f, g ) → g∘f , and all attempts to change this notation have, so far, failed).
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. …
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, machinery such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might be, used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.
The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for …
The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.
A characterization is given of those bicategories which are biequivalent to categories of modules for some suitable base. These bicategories are the correct (non elementary) notion of cosmos, which is …
A characterization is given of those bicategories which are biequivalent to categories of modules for some suitable base. These bicategories are the correct (non elementary) notion of cosmos, which is shown to be closed under several basic constructions.
The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting …
The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we show that this construction is unique up to an invertible scalar. Due to this fact, Hopf-Frobenius algebras provide two canonical notions of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but has a much simpler presentation.
This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories.Any small …
This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories.Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve.We prove that the geometric realizations of all of these 'nerves of the tricategory' are homotopy equivalent.By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence.Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space.With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3-types.
Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a …
Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras, and Lie algebras.
We investigate when a weak Hopf algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Frobenius. We show this is not always true, but …
We investigate when a weak Hopf algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Frobenius. We show this is not always true, but it is true if the semisimple base algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has all its matrix blocks of the same dimension. However, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a semisimple algebra not having this property, there is a weak Hopf algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with base <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not Frobenius (and consequently, it is not Frobenius “over” <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> either). Moreover, we give a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of a weak Hopf algebra.
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras …
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter–Drinfeld modules over Hopf group-coalgebras using the center construction.
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We …
We introduce the notion of an oplax Hopf monoid in any braided monoidal bicategory, generalizing that of a Hopf monoid in a braided monoidal category in an appropriate way. We show that Hopf V-categories introduced in [BCV16] are a particular type of oplax Hopf monoids in the monoidal bicategory Span|V described in [B\oh17]. Finally, we introduce Frobenius V-categories as the Frobenius objects in the same monoidal bicategory.
We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the related (opmonoidal) monad is a …
We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the related (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.
This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of …
This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students. Titles in this series are co-published with the Centre de Recherches Mathematiques.|This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students. Titles in this series are co-published with the Centre de Recherches Mathematiques.