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Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in set-theoretic terms. This paper examines the analogous situation for structures described in category-theoretic terms, where symmetry is expressed as the action of a 2-group G (equivalently, a categorical group) on a category C. It describes the construction of a transformation groupoid in diagrammatic terms, and considers this construction internal to Cat, the category of categories. The result is a double category C//G which describes the local symmetries of C. We define this and describe some of its structure, with the adjoint action of G on itself as a guiding example.
Keywords: 2-group, categorical group, crossed module, action, double category, adjoint action
Jeffrey C. Morton; Roger Picken. Transformation double categories associated to 2-group actions. Theory and applications of categories, Tome 30 (2015), pp. 1429-1468. http://geodesic.mathdoc.fr/item/TAC_2015_30_a42/
@article{TAC_2015_30_a42,
author = {Jeffrey C. Morton and Roger Picken},
title = {Transformation double categories associated to 2-group actions},
journal = {Theory and applications of categories},
pages = {1429--1468},
year = {2015},
volume = {30},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2015_30_a42/}
}