A Skew-Duoidal Eckmann-Hilton Argument and Quantum Categories
نویسندگان
چکیده
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.
منابع مشابه
The Eckman - Hilton argument and higher operads
To the memory of my father. Abstract The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppo...
متن کاملTannaka Duality and Convolution for Duoidal Categories
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 expresses Tannaka duality. The approach...
متن کاملSkew Monoidales, Skew Warpings and Quantum Categories
Kornel Szlachányi [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...
متن کاملQuillen Spectral Sequences in Homology and Rational Homotopy of Cofibrations
We construct Quillen type spectral sequences in homology and rational homotopy for coobration sequences which are Eckmann-Hilton dual to analogous ones for bration sequences. These spectral sequences are constructed by direct ltrations of the Adams cobar construction. We also prove various collapsing theorems generalizing results of Clark and Smith in the case of a wedge of 1-connected nicely p...
متن کاملWeak units and homotopy 3 - types
We show that every braided monoidal category arises as End(I) for a weak unit I in an otherwise completely strict monoidal 2-category. This implies a version of Simpson’s weak-unit conjecture in dimension 3, namely that one-object 3-groupoids that are strict in all respects, except that the object has only weak identity arrows, can model all connected, simply connected homotopy 3-types. The pro...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Applied Categorical Structures
دوره 22 شماره
صفحات -
تاریخ انتشار 2014