Crossed semimodules of categories and Schreier 2-categories. (English) Zbl 1444.18011

Summary: The main idea of this paper is to introduce the notion of a Schreier 2-category and of a crossed semimodule over categories and to prove the categorical equivalence between their categories.


18C40 Structured objects in a category (group objects, etc.)
18B99 Special categories
18M05 Monoidal categories, symmetric monoidal categories
20L99 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
Full Text: DOI Euclid


[1] Baez, J.C., Baratin, A., Freidel, L. and Wise, D.K., Infinite-Dimensional Representations of 2-Groups, Memoirs of the American Mathematical Society, Volume 219, Number 1032, (2012). · Zbl 1342.18008
[2] Baez, J.C., Lauda, A.D., Higher-dimensional algebra V: 2-groups, Theory Appl. Categ. 12, 423-491 (2004). · Zbl 1056.18002
[3] \(B \acute{e}\) nabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar Lecture Notes in Mathematics Volume 47, pp 1-77 (1967).
[4] Brown, R. and Higgins, P.J., Crossed Complexes and non-Abelian Extensions, Georgian Mathematical Journal, Vol. 962, No. 6, 39-50 (1981).
[5] Brown, R. and Higgins, P.J., Tensor Products and Homotopies for \(\omega-\) groupoids and crossed complexes, J. Pure and Appl. Algebra, 47, 1-33 (1987). · Zbl 0621.55009
[6] Brown, R. and Icen, I, Homotopies and Automorphisms of Crossed Module Over Groupoids, Appl. Categorical Structure, 11, 185-206 (2003). · Zbl 1029.20025
[7] Brown, R., Spencer, C.B., \( \mathcal{G} \)-groupoids, crossed modules and the fundamental groupoid of a topological group. Proc. Konn. Ned. Akad. 1976; 79: 296-302. · Zbl 0333.55011
[8] Gursoy, M.H., Icen, I. and Ozcan, A.F., The Equivalence of Topological 2-Groupoids an Topological Crossed Modules, Algebras Groups and Geometries, 22, 447-456 (2005). · Zbl 1114.22004
[9] Icen, I., The Equivalence of 2-Groupoids an Crossed Modules, Commun. Fac. Sci. Univ. Ank. Series A1, Vol:49, 39-48 (2000). · Zbl 0994.18003
[10] Maclane, S., Categories for the Working Mathematician, Graduate Text in Mathematics, Volume 5. Springer-Verlag, New York (1971). · Zbl 0705.18001
[11] Noohi, B., Notes on 2-Groupoids, 2-Groups and Crossed Modules, Homology Homotopy Appl. Volume 9, Number 1, 75-106 (2007). · Zbl 1221.18002
[12] Patchkoria, A., Crossed Semimodules and Schreier Internal Categories In The Category of Monoids, Georgian Mathematical Journal, Vol. 5, No. 6, 575-581 (1998). · Zbl 0915.18002
[13] Porter, T., Crossed Modules in Cat and a Brown-Spencer Theorem for 2-Categories, Cahiers de Topologie et Geometrie Differentielle Categoriques, Vol. XXVI-4 (1985). · Zbl 0575.18006
[14] Porter, T., Extensions, Crossed Modules and Internal Categories in Categories of Groups With Operations, Proceedings of the Edinburgh Mathematical Society 30, 371-381 (1987). · Zbl 0595.18006
[15] Temel, S., Topological Crossed Semimodules and Schreier Internal Categories in the Category of Topological Monoids, Gazi University Journal of Science, 29(4):915-921 (2016).
[16] Whitehead, J.H.C., Combinatorial homotopy II, Bull. Amer. Math. Soc. 55,453-496(1949). · Zbl 0040.38801
[17] Whitehead, J.H.C., \em Note on a previous paper entitled “On adding relations to homotopy group”, Ann. Math. 47, 806Â-810 (1946). · Zbl 0060.41104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.