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Axioms 2015, 4(1), 32-70; doi:10.3390/axioms4010032

Azumaya Monads and Comonads

1
Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6, Tamarashvili Str., Tbilisi Centre for Mathematical Sciences, Chavchavadze Ave. 75, 3/35, Tbilisi 0177, Georgia
2
Department of Mathematics, Heinrich Heine University, Düsseldorf 40225, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Florin Nichita
Received: 1 October 2014 / Revised: 11 December 2014 / Accepted: 4 January 2015 / Published: 19 January 2015
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)
View Full-Text   |   Download PDF [435 KB, uploaded 19 January 2015]

Abstract

The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \(\mathbb{A}\) by considering a monad \((F,m,e)\) on \(\mathbb{A}\) endowed with a distributive law \(\lambda: FF\to FF\) satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad \((F^\lambda,m\cdot \lambda,e)\) and a monad structure on \(FF^\lambda\). The quadruple \((F,m,e,\lambda)\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \(\mathbb{A}\) and the category of \(FF^\lambda\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \(F\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\(,\otimes,I,\tau)\), for any V-algebra \(A\), the braiding induces a BD-law \(\tau_{A,A}:A\otimes A\to A\otimes A\), and \(A\) is called left (right) Azumaya, provided the monad \(A\otimes-\) (resp. \(-\otimes A\)) is Azumaya. If \(\tau\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide. View Full-Text
Keywords: Azumaya algebras, category equivalences, monoidal categories, (co)monads Azumaya algebras, category equivalences, monoidal categories, (co)monads
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Mesablishvili, B.; Wisbauer, R. Azumaya Monads and Comonads. Axioms 2015, 4, 32-70.

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