Furber and Jacobs have shown in their study of quantum computation that the category of commutative C∗-algebras and PU-maps (positive linear maps which preserve the unit) is isomorphic to the Kleisli category of a comonad on the category of commutative C∗-algebras with MIU-maps (linear maps which preserve multiplication, involution and unit). [3] In this paper, we prove a non-commutative varian...

1. DERIVED GEOMETRY WITH L∞ ALGEBRAS We are interested in studying formal derived moduli problems, as an orienting remark recall that particularly nice simplicial sets are those that are Kan complexes and that the nerve of a category C is Kan complex if and only if C is a groupoid. Consider the following progression. • Schemes: Functors from commutative algebras to sets; • Stacks: Functors from...

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local

We continue a previous study on Γ-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any Z-graded vertex algebra V and a positive integer N , the category of V -modules is naturally isomorphic to the category of quasimodules of a certain type for V ⊗N . We also study certain generalizations of twisted affine Lie algebras and we relate suc...

The notion of central series for groups with action on itself is introduced. An analogue of Witt’s construction is given for such groups. A certain condition is found for the action and the corresponding category is defined. It is proved that the above construction defines a functor from this category to the category of Lie–Leibniz algebras and in particular to Leibniz algebras; also the restri...

By defining a closure operator on effective equivalence relations in a regular category C, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories of C, on the model of the closure operators on kernels in homological categories [5]. When C is an exact Goursat category [6], this correspondence restricts to a bijection bet...

The cluster category was introduced in (BMRRT, 2006) and also in (CCS1, 2006) for type A, as a categorical model to understand better the cluster algebras of Fomin and Zelevinsky (FZ, 2002). It is a quotient of the bounded derived category Db(modA) of the finitely generated modules over a finite dimensional hereditary algebra A. It was then natural to consider the endomorphism algebras of tilti...

We show that Turaev’s group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier’s version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over ...

The category psBCI of pseudo-BCI-algebras and homomorphisms between them is investigated. It is also shown that the category psBCIp of p-semisimple pseudo-BCI-algebras and homomorphisms between them is a reflective subcategory of psBCI.

A group completion functor Q is constructed in the category of algebras in simplicial sets over a cofibrant En-operad M. It is shown that Q defines a Bousfield–Friedlander simplicial model category on M-algebras.