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We define exact sequences in the enchilada category of $C^*$-algebras and correspondences, and prove that the reduced-crossed-product functor is not exact for the enchilada categories. Our motivation was to determine whether we can have a better understanding of the Baum-Connes conjecture by using enchilada categories. Along the way we prove numerous results showing that the enchilada category is rather strange.
We generalize the definition of an exact sequence of tensor categories due to Brugui`eres and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this n
We introduce the concept of an infinite cochain sequence and initiate a theory of homological algebra for them. We show how these sequences simplify and improve the construction of infinite coclass families (as introduced by Eick and Leedham-Green) a
In the context of A. Connes spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. A. Connes construction of spectral triples for group algebras is a covarian
We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.
We construct several smooth finite element spaces defined on three--dimensional Worsey--Farin splits. In particular, we construct $C^1$, $H^1(curl)$, and $H^1$-conforming finite element spaces and show the discrete spaces satisfy local exactness prop