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Although the notion of crossed groups was originally introduced only in thesimplicial case, the definition makes sense in the other categories. Forinstance, Batanin and Markl studied crossed interval groups to investigatesymmetries on the Hochschild cohomology in view of operads. The aim of thispaper is to make a comprehensive understanding of crossed groups for arbitrarybase categories. In particular, we focus on the local presentability of thecategory of crossed groups, monadicity, and the basechange theorem alongcertain sorts of functors. The paper also contains the classification ofcrossed interval groups, which Batanin and Markl concerned about.

Limits and colimits of crossed groups