Let U(g) denote the universal enveloping algebra of a Lie algebra g. We show the existence of a ribbon algebra structure in a particular deformation of U(g) which leads to a symmetric pre-monoidal category of U(g)-modules.

We develop a theory of double clubs which extends Kelly’s theory of clubs to the pseudo double categories of Paré and Grandis. We then show that the club for symmetric strict monoidal categories onCat extends to a ‘double club’ on the pseudo double category Cat of ‘categories, functors, profunctors and transformations’.

We develop some aspects of the homological algebra persistence modules, in both one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. consider module sheaf tensor product Hom bifunctors well their derived functors, Tor Ext, give explicit computations for interval a classification injective, projective, flat state Kunneth theo...

We prove Steinebrunner's conjecture on the biequivalence between (colored) properads and labelled cospan categories. The main part of work is to establish a 1-categorical, strict version conjecture, showing that category equivalent categories via symmetric monoidal envelope functor.

A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is introduced on the restriction of this category to isometries, which trace is carried over to directed quantum Turing machines as monoidal automata. The JoyalStreet-...

We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists a strong traced functor into FinVectk which disti...

The 2-categorical approach to quantum information is now well-developed in its basic aspects [4, 12, 14]. The central aim is to use the structure of a symmetric monoidal 2-category to describe quantum procedures in an abstract way, such that the ordinary versions of these procedures are recovered when we apply the formalism in 2Hilb, the symmetric monoidal 2-category of 2-Hilbert spaces [3]. Th...

We obtain combinatorial model categories of parametrised spectra, together with systems base change Quillen adjunctions associated to maps parameter spaces. work simplicial objects and use Hovey's sequential symmetric stabilisation machines. By means a Grothendieck construction for categories, we produce controlling the totality stable homotopy theory. The global category spectra is equipped mo...

Graded monoidal categories were introduced by Frohlich and C.T.C. ̈ Wall in 8 , where they presented a suitable abstract setting to study the Brauer group in equivariant situations. This paper is concerned with the analysis and classification of these graded monoidal categories, following a parallel treatment to that made in 2 for the non-monoidal case. In any graded monoidal category, its 1-com...