The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functor...

Homotopy theory studies topological spaces up to homotopy, so it must study the functor π∗ : Top → Π−algebras. Passage to the homotopy category is declaring π∗-isomorphisms to be isomorphisms. Because this functor is difficult to compute, one way to do homotopy theory is to study simpler functors which also map to graded abelian categories, e.g. πQ ∗ , H∗(−;Z), K∗, or more generally E∗ for E so...

Recently, Turaev [19, 20] (see also Le and Turaev [8] and Virelizier [23]) generalized the notion of a TQFT and Reshetikhin-Turaev invariants to the case of 3-manifolds endowed with a homotopy classes of maps to K(π, 1), where π is a group. One of the key points in [19] is the notion of a crossed Hopf π-coalgebra, here called a Turaev coalgebra or, briefly, a T-coalgebra (see Section 1). As one...

We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*-algebras. The other sees quantales as algebras of abstract experiments on physical or computational systems, and has been applied to t...

These notes were written to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in Fall 2008 at the University of Chicago. We first introduce orthogonal factorization systems, give a few examples, and prove some basic theorems. Next, we turn to weak factorization systems, which play an important role in the theory of model categories, a connection which we make expli...

Prof. Boris I. Plotkin [1, 2] drew attention to the question when an equivalence between two categories is isomorphic as a functor to an isomorphism between them. It turns out that it is quite important for universal algebraical geometry and concerns mainly the categories Θ0(X) of free universal algebras of some variety Θ free generated by finite subsets of X. In the paper, a complete answer to...