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Register a new userCohomology for small categories: $k$-graphs and groupoids
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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 $G_Lambda$.
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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 finite depth finite index connected hyperfinite $rm II_1$ multifactor inclusions $Asubset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popas uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $rm II_1$ multifactor inclusions.
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 whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gelfand-Naimark representation theorem for Krein C*-algebras and Krein 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 category $mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $mathcal{C}$ and $mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $mathcal{C}$ and $mathcal{D}$ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: [ Big(,,mathcal{C},,simeq_{text{Morita}},,mathcal{D},,Big) qquadLongrightarrowqquad Big(,,mathcal{C},,simeq_{text{tensor}},,mathcal{D},,Big). ] This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional $rm C^*$-algebras are isomorphic von Neumann algebras, provided the representations are `big enough. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $rm C^*$-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $mathcal{C}tooperatorname{Bim}(R)$ automatically respect these positive structures.
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 independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
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 over $ C^* $-tensor categories exist without any finiteness assumptions.
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