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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.
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 proposa
Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category $mathcal{D}$ is an infinite loop space, the zeroth space of the $K$-theory spectrum $Kmathcal{D}$. This spectrum
We classify various types of graded extensions of a finite braided tensor category $cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $cal B$ by a finite group $A$ correspond to braided monoidal
We analyze the action of the Brauer-Picard group of a pointed fusion category on the set of Lagrangian subcategories of its center. Using this action we compute the Brauer-Picard groups of pointed fusion categories associated to several classical fin
Each object of any abelian model category has a canonical resolution as described in this article. When the model structure is hereditary we show how morphism sets in the associated homotopy category may be realized as cohomology groups computed from