Representation theory of 2-groups on finite dimensional 2-vector spaces

Abstract

In this paper, the 2-category ${\mathfrak{Rep}}_{{2Mat}_{\mathbb{C}}}(\mathbb{G})$ of (weak) representations of an arbitrary (weak) 2-group $\mathbb{G}$ on (some version of) Kapranov and Voevodsky's 2-category of (complex) 2-vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants ${\pi }_0(\mathbb{G})$, ${\pi }_1(\mathbb{G})$ and $[\alpha ]\in H^3({\pi }_0(\mathbb{G}),{\pi }_1(\mathbb{G}))$ classifying $\mathbb{G}$. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain the the {\it monoidal} category of linear representations (more generally, the category of $[z]$-projective representations, for any given cohomology class $[z]\in H^2({\pi }_0(\mathbb{G}),{\mathbb{C}}^{\ast }))ofthefirsthomotopygroup$π_0(\mathbb{G})$aswellasitscategoryofrepresentationsonfinitesetsbothlivein$\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$,thefirstasthemonoidalcategoryofendomorphismsofthetrivialrepresentation(moregenerally,asthecategoryofmorphismsbetweensuitable1-dimensionalrepresentations)andthesecondasasubcategoryofthehomotopycategoryof$\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$.

Completely new version. In particular, weak representation theory of arbitrary weak 2-groups is treated

References in corpus (3)

• Higher-Dimensional Algebra V: 2-Groups
• Categorical representations of categorical groups
• Group Objects and Internal Categories

Cited by in corpus (6)

• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Infinite-Dimensional Representations of 2-Groups
• On weak maps between 2-groups
• AQFT from n-functorial QFT
• 2-Groups, trialgebras and their Hopf categories of representations
• The geometry of unitary 2-representations of finite groups and their 2-characters