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Given a higher-rank graph $Lambda$, we investigate the relationship between the cohomology of $Lambda$ and the cohomology of the associated groupoid $G_Lambda$. We define an exact functor between the abelian category of right modules over a higher-rank graph $Lambda$ and the category of $G_Lambda$-sheaves, where $G_Lambda$ is the path groupoid of $Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $Lambda$ with coefficients in a right $Lambda$-module, and the continuous cocycle cohomology of $G_Lambda$ with values in the corresponding $G_Lambda$-sheaf, into the sheaf cohomology of $G_Lambda$.
We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify fi
C*-categories are essentially norm-closed *-categories of bounded linear operators between Hilbert spaces. The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $mathcal{C}$ of a fully faithful representation $mathcal{C}tooperatorname{Bim}(R)$ of a unitary fusion ca
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Gamma$-objects in 2-categories. In the course of the proof we establish strictfication results of ind
We discuss a number of general constructions concerning additive $ C^* $-categories, focussing in particular on establishing the existence of bicolimits. As an illustration of our results we show that balanced tensor products of module categories ove