defined on n-tuples of commuting elements of G and invariant under simultaneous conjugation. Thus, in [HKR00] the study of nth Morava E-theory at a prime p leads to χ(g1, ..., gn) defined on gi of p-power order. On the other hand, the construction of equivariant elliptic cohomology in [Dev96] (which corresponds to n = 2) uses functions χ(g1, g2) as above but without restrictions on the order. T...

G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner’s approach to deriving labelled bisimulation congruences from reduction systems. This paper develops the theory of GRPOs further, arguing that they provide a simple and powerful basis towards a comprehensive solution. As an example, we construct GRPOs in a category of ‘bunches and wi...

We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-pushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that sufficiently many GRPOs exist...

Derived categories possibly give a new significant invariant for algebraic varieties. In particular, when the caninical line bundle is trivial, there are varieties which are not birationally equivalent, but have equivalent derived categories. The most typical example can be found in the original paper [Mu]. On the other hand, for such varieties, it is hoped that the birationally equivalence sho...

We introduce the class of isotypic 2-representations for finitary 2categories and the notion of inflation of 2-representations. Under some natural assumptions we show that isotypic 2-representations are equivalent to inflations of cell 2-representations.

We study connections between additive and abelian 2-representations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell 2-representation. We also describe, in detail, examples of fiat 2-categories associated to sl2categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.