Korean J. Math.  Vol 27, No 2 (2019)  pp.535-545
DOI: https://doi.org/10.11568/kjm.2019.27.2.535

Crossed semimodules and cat$^{\bf 1}$-monoids

Sedat Temel


The main idea of this paper is to introduce the notion of cat$^{\bf{1}}$-monoids and to prove that the category of crossed semimodules $ \mathcal{C} = (A,B,\partial) $ where $ A $ is a group is equivalent to the category of cat$^{\bf{1}}$-monoids. This is a generalization of the well known equivalence between category of cat$^{\bf{1}}$-groups and that of crossed modules over groups.


Crossed module, crossed semimodule, cat$^1$-group, cat$^1$-monoid

Subject classification

20L05, 18D05, 18D35, 20J15


Full Text:



Akız, H.F., Alemdar, N., Mucuk, O. and S ̧ahan, T., Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Mathematical Journal 20 (2) (2013), 223–238. (Google Scholar)

Baez, J.C. and Lauda, A.D., Higher Dimensional Algebra V: 2-Groups, Theory and Applications of Categories 12 (14) (2004), 423–491. (Google Scholar)

Baez J.C. and Stevenson D. The classifying space of a topological 2-group, Al- gebraic Topology Abel Symposia; 4: pp 1-31 (2009). (Google Scholar)

Baez, J.C., Baratin, A., Freidel, L. and Wise, D.K., Infinite-Dimensional Repre- sentations of 2-Groups, Memoirs of the American Mathematical Society, Volume 219, Number 1032, (2012). (Google Scholar)

Brown, R., Groupoids and crossed objects in algebraic topology, Homology Homotopy Appl. 1 (1999), 1–78. (Google Scholar)

Brown, R., Higher dimensional group theory. In: Low Dimensional Topology, London Math. Soc. Lect. Notes, 48, pp. 215238. Cambridge Univ. Press, (1982) (Google Scholar)

Brown, R. and Huebschmann, J., Identities among relations. In: Low Dimentional Topology, London Math. Soc. Lect. Notes, 48, pp. 153–202. Cambridge Univ. Press (1982). (Google Scholar)

Brown, R. and Mucuk, O., Covering groups of non-connected topological groups revisited, Math. Proc. Camb. Phil. Soc. 115 (1994), 97–110. (Google Scholar)

Brown, R. and Spencer, C.B., G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Konn. Ned. Akad. 79 (1976), 296–302. (Google Scholar)

Brown, R., Higgins, P. J. and Sivera, R., Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids, European Mathematical Society Tracts in Mathematics 15 (2011). (Google Scholar)

Brown, R and Huebschmann, J., Identities among relations, pages 153-202. London Mathematical Society Lecture Note Series. Cambridge University Press, (1982). (Google Scholar)

Huebschmann, J., Crossed n-fold extensions of groups and cohomology, Comment. Math. Helvetici 55 (1980), 302–314. (Google Scholar)

Loday, J.-L., Cohomologie et groupes de Steinberg relatifs, J. Algebra 54 (1978), 178–202. (Google Scholar)

Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J.Pure Appl. Algebra 24 (2) (1982), 179–202 . (Google Scholar)

Lue, A.S.T., Cohomology of groups relative to a variety, J. Algebra 69 (1981), 155–174. (Google Scholar)

Maclane, S., Categories for the Working Mathematician, Graduate Text in Mathematics, Volume 5. Springer-Verlag, New York (1971). (Google Scholar)


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