Group Objects and Internal Categories

Abstract

Algebraic structures such as monoids, groups, and categories can be formulated within a category using commutative diagrams. In many common categories these reduce to familiar cases. In particular, group objects in Grp are abelian groups, while internal categories in Grp are equivalent both to group objects in Cat and to crossed modules of groups. In this exposition we give an elementary introduction to some of the key concepts in this area.

12 pages, expository article

References in corpus (1)

• Computation and Homotopical Applications of Induced Crossed Modules

Cited by in corpus (25)

• An Invitation to Higher Gauge Theory
• Integrating L-infinity algebras
• Higher Gauge Theory: 2-Connections on 2-Bundles
• Higher Yang-Mills Theory
• Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics
• Infinite-Dimensional Representations of 2-Groups
• Pseudo Algebras and Pseudo Double Categories
• A supergeometric approach to Poisson reduction
• Coherence for monoidal $G$-categories and braided $G$-crossed categories
• 2-Groups, trialgebras and their Hopf categories of representations
• Strict 2-Groups are Crossed Modules
• Representation theory of 2-groups on finite dimensional 2-vector spaces
• Categorified central extensions, étale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras
• Sur l'existence d'une catégorie ayant une matrice strictement positive donnée
• Twisted-Product Categorical Bundles
• Twisted actions of categorical groups
• Integrating central extensions of Lie algebras via Lie 2-groups
• On the existence of a category with a given matrix
• On state sums, internalisation and unification
• Classification des catégories associées à la matrice des coefficients 2 d'ordre donnée
• Grothendieck-topological Group Objects
• Current Groups and the Hamiltonian Anomaly
• (Weak) incidence bialgebras of monoidal categories
• Path space connections and categorical geometry
• Strict Gr-categories and applications on classification of extensions of groups of the type of a crossed module