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- Brian Day
- 2006

The purpose of the present paper is to develop in further detail the remarks, concerning the relationship of Kan functor extensions to closed structures on functor categories, made in "Enriched functor categories" | 1] §9. It is assumed that the reader is familiar with the basic results of closed category theory, including the representation theorem. Apart… (More)

- Brian Day
- 2003

A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term "quantum category"in a braided monoidal category with equalizers distributed over by tensoring with an object. The definition of antipode for a bialgebroid… (More)

- Brian Day, Paddy McCrudden, Ross Street
- Applied Categorical Structures
- 2003

- BRIAN DAY
- 1995

Strong promonoidal functors are defined. Left Kan extension (also called " existential quantification ") along a strong promonoidal functor is shown to be a strong monoidal functor. A construction for the free monoidal category on a promonoidal category is provided. A Fourier-like transform of presheaves is defined and shown to take convolution product to… (More)

- THOMAS BOOKER, Brian Day
- 2013

Given a horizontal monoid M in a duoidal category F , we examine the relationship between bimonoid structures on M and monoidal structures on the category F * M of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the so-called Tannaka adjunction; that is, an adjunction inducing the equivalence which… (More)

- Brian A. Day
- 2000

- Brian Day, Craig Pastro
- 2008

It is well known that strong monoidal functors preserve duals. In this short note we show that a weaker version of functor, which we call " Frobenius monoidal " , is sufficient. Further properties of Frobenius monoidal functors are developed. Throughout suppose that A and B are strict 1 monoidal categories. Definition 1. A Frobenius monoidal functor is a… (More)

- BRIAN DAY, CRAIG PASTRO
- 2008

We show that the (co)endomorphism algebra of a sufficiently separable " fibre " functor into Vect k , for k a field of characteristic 0, has the structure of what we call a " unital " von Neumann core in Vect k. For Vect k , this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a… (More)

- Brian Day
- 2009

We discuss an inversion property of the fusion map associated to many semibialgebras. Please note that a characterisation of Hopf k-algebras has been added at the end of this version. be a symmetric (or just braided) monoidal category. A Von Neumann " core " in C is firstly a semibialgebra in C, that is, an object A in C with an associative multiplication:… (More)

- Brian Day, STEPHEN LACK
- 2015

This paper extends the Day Reection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore algebras for a comonad.