Dimitri Chikhladze’s research while affiliated with Macquarie University and other places

In \cite{LS14} the analogy between the Kleisli construction and the construction of "warping a skew monoidale category" in the sense of \cite{LS12} was outlined. In this note we present the same work in a slightly more formal way.

Working in the framework of $(\mathbb {T},\textbf {V})$ -categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad $\mathbb {T}$ , we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for $(\mathbb {T},\textbf {V})$ -categories.

Generalized operads, also called generalized multicategories and T-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors in different contexts, with examples including symmetric multicategories, topological spaces, globular operads and Lawvere theories. In this paper we study functoriality of the Kleisli construction, and correspondingly that of generalized operads. Motivated by this problem we develop a lax version of the formal theory of monads, and study its connection to bicategorical structures.

Working in the framework of $(T, V)$-categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad T, we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for $(T, V)$-categories.

Generalized multicategories, also called T-monoids, are well known class of mathematical structures, which include diverse set of examples. In this paper we construct a generalization of the adjunction between strict monoidal categories and multicategories, where the latter are replaced by T-monoids. To do this we introduce lax monads in a 3-category, and establish their relationship with equipments, which are bicategory like structures appropriate for the generalized multicategory theory.

A weak bialgebra is known to be a special case of a bialgebroid. In this paper we study the relationship of this fact with the Tannaka theory of bialgebroids as developed in [4]. We obtain a Tannaka representation theorem with respect to a separable Frobenius fiber functor. Comment: 7 pages

There are various generalizations of bialgebras to their ''many object'' versions, such as quantum categories, bialgebroids and weak bialgebras. These can also be thought of as quantum analogues of small categories. In this paper we study modules over these structures, which are quantum analogues of profunctors (also called distributors) between small categories. Comment: 23 pages

Alain Bruguieres, in his talk [1], announced his work [2] with Alexis Virelizier and the second author which dealt with lifting closed structure on a monoidal category to the category of Eilenberg-Moore algebras for an opmonoidal monad. Our purpose here is to generalize that work to the context internal to an autonomous monoidal bicategory. The result then applies to quantum categories and bialgebroids.

Quantum categories were introduced in [5] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set of axioms close to the definitions of a bialgebroid in the Hopf algebraic literature. We introduce notions of functor and natural transformation for quantum categories and consider various constructions on quantum structures. © Dimitri Chikhladze, 2011. Permission to copy for private use granted.