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We provide definitions for strict involutive higher categories (a vertical categorification of dagger categories), strict higher C*-categories and higher Fell bundles (over arbitrary involutive higher topological categories). We put forward a proposal for a relaxed form of the exchange property for higher (C*)-categories that avoids the Eckmann-Hilton collapse and hence allows the construction of explicit non-trivial non-commutative examples arising from the study of hypermatrices and hyper-C*-algebras, here defined. Alternatives to the usual globular and cubical settings for strict higher categories are also explored. Applications of these non-commutative higher C*-categories are envisaged in the study of morphisms in non-commutative geometry and in the algebraic formulation of relational quantum theory.
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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.
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.
In this paper, we state the notion of morphisms in the category of abelian crossed modules and prove that this category is equivalent to the category of strict Picard categories and regular symmetric monoidal functors. The theory of obstructions for symmetric monoidal functors and symmetric cohomology groups are applied to show a treatment of the group extension problem of the type of an abelian crossed module.
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.
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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