559 research outputs found
Skew-closed categories
Spurred by the new examples found by Kornel Szlach\'anyi of a form of lax
monoidal category, the author felt the time ripe to publish a reworking of
Eilenberg-Kelly's original paper on closed categories appropriate to the laxer
context. The new examples are connected with bialgebroids. With Stephen Lack,
we have also used the concept to give an alternative definition of quantum
category and quantum groupoid. Szlach\'anyi has called the lax notion {\em skew
monoidal}. This paper defines {\em skew closed category}, proves Yoneda lemmas
for categories enriched over such, and looks at closed cocompletion.Comment: Version 2 corrects a mistake in axiom (2.4) noticed by Ignacio Lopez
Franco. Only the corrected axiom was used later in the paper so no other
consequential change was needed. A few obvious typos have been corrected.
Some material on weighted colimits, composite modules and skew-promonoidal
categories has been added. Version 3 adds Example 23 and corrects a few
typos.
Vector product and composition algebras in braided monoidal additive categories
This is an account of some work of Markus Rost and his students Dominik Boos
and Susanne Maurer. We adapt it to the braided monoidal setting.Comment: 23 page
On monads and warpings
We explain the sense in which a warping on a monoidal category is the same as
a pseudomonad on the corresponding one-object bicategory, and we describe
extensions of this to the setting of skew monoidal categories: these are a
generalization of monoidal categories in which the associativity and unit maps
are not required to be invertible. Our analysis leads us to describe a
normalization process for skew monoidal categories, which produces a universal
skew monoidal category for which the right unit map is invertible.Comment: 15 pages. Version 2: revised based on a very helpful report from the
referee. To appear in the Cahiers de Topologie and Geometrie Differentielle
Categorique
Triangulations, orientals, and skew monoidal categories
A concrete model of the free skew-monoidal category Fsk on a single
generating object is obtained. The situation is clubbable in the sense of G.M.
Kelly, so this allows a description of the free skew-monoidal category on any
category. As the objects of Fsk are meaningfully bracketed words in the skew
unit I and the generating object X, it is necessary to examine bracketings and
to find the appropriate kinds of morphisms between them. This leads us to
relationships between triangulations of polygons, the Tamari lattice, left and
right bracketing functions, and the orientals. A consequence of our description
of Fsk is a coherence theorem asserting the existence of a strictly
structure-preserving faithful functor from Fsk to the skew-monoidal category of
finite non-empty ordinals and first-element-and-order-preserving functions.
This in turn provides a complete solution to the word problem for skew monoidal
categories.Comment: 48 page
Closed categories, star-autonomy, and monoidal comonads
This paper determines what structure is needed for internal homs in a
monoidal category C to be liftable to the category C^G of Eilenberg-Moore
coalgebras for a monoidal comonad G on C. We apply this to lift star-autonomy
with the view to recasting the definition of quantum groupoid.Comment: 25 page
Torsors, herds and flocks
This paper presents non-commutative and structural notions of torsor. The two
are related by the machinery of Tannaka-Krein duality
A skew-duoidal Eckmann-Hilton argument and quantum categories
A general result relating skew monoidal structures and monads is proved. This
is applied to quantum categories and bialgebroids. Ordinary categories are
monads in the bicategory whose morphisms are spans between sets. Quantum
categories were originally defined as monoidal comonads on endomorphism objects
in a particular monoidal bicategory M. Then they were shown also to be skew
monoidal structures (with an appropriate unit) on objects in M. Now we see in
what kind of M quantum categories are merely monads.Comment: 14 pages, dedicated to George Janelidze on the occasion of his 60th
birthday; v2 final version, 15 pages, to appear in Applied Categorical
Structure
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