In this paper, starting with a simply laced root system, we define a triangulated category which we call the m-cluster category, and we show that it encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. For Φ any root sys...

Abstract. We introduce the category of set-theoretic representations of a matched pair of groupoids. This is a monoidal category endowed with a monoidal functor fl to the category of quivers over the common base of the groupoids in the matched pair. We study monoidal functors between two such categories of representations which preserve the functor fl. We show that the centralizer of such a mon...

We extend R. Fernandes’ construction of the secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism. In [4] R. Fernandes has constructed a sequence of secondary characteristic classes of a Li...

We consider projectivity and injectivity of Hilbert C*-modules in the categories of Hilbert C*-(bi-)modules over a fixed C*-algebra of coefficients (and another fixed C*-algebra represented as bounded module operators) and bounded (bi-)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characteri...

We introduce a new concept in the area of formal logic: axioms for model morphisms. We work in the setting of specification languages that define the semantics of a theory as a category of models. While it is routine to use axioms to specify the class of models of a theory, there has so far been no analogue to systematically specify the morphisms between these models. This leads to subtle probl...

Let X and A be sets and α : X → A a map between them. We call a map μ : X ×X ×X → A an approximate Mal’tsev operation with approximation α, if it satisfies μ(x, y, y) = α(x) = μ(y, y, x) for all x, y ∈ X. Note that if A = X and the approximation α is an identity map, then μ becomes an ordinary Mal’tsev operation. We prove the following two characterization theorems: a category X is a Mal’tsev c...

Abstract We transfer several elementary geometric properties of rigid-analytic spaces to the world adic spaces, more precisely category which are locally (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and Stein factorization theorem. Most notably, we show that morphism in our is automatically open if target nor...