Crossed squares, crossed modules over groupoids and cat$^{\bf {1-2}}-$groupoids

Document Type: Research Paper


Department of Mathematics, Faculty of Arts and Science, Recep Tayyip Erdogan University, Rize, Turkey.


The aim of this paper is to introduce the notion of cat$^{\bf {1}}-$groupoids which are the groupoid version of cat$^{\bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{\bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{\bf {2}}-$groupoids, and then we show their categories are equivalent. These equivalences enable us to obtain more examples of groupoids.


[1] Akiz, H.F., Alemdar, N., Mucuk, O., Sahan, T., Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J. 20(2) (2013), 223-238.
[2] Ataseven, C ., Relations among higher order crossed modules over groupoids, Konu- ralp J. Math. 4(1) (2016), 282-290.
[3] Baez, J.C. and Lauda, A.D., Higher dimensional algebra V: 2-groups, Theory Appl. Categ. 12(14) (2004), 423-491.
[4] Brown, R., "Topology and Groupoids", BookSurge LLC, 2006.
[5] Brown, R. and Spencer, C.B., G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indag. Math. (N.S.) 79(4) (1976), 296-302.
[6] Brown, R., Higgins, P.J., Sivera, R., "Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids", Eur. Math. Soc. Tracts in Math. 15, 2011.
[7] Brown, R. and Higgins, P.J., Tensor products and homotopies for !-groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), 1-33.
[8] Brown R. and Higgins P.J., Crossed complexes and non-abelian extensions, In: Cat- egory Theory. Lecture Notes in Math. 962, Springer, 1982.
[9] Brown, R. and Icen, I., Homotopies and Automorphisms of Crossed Module Over Groupoids, Appl. Categ. Structures 11 (2003), 185-206.
[10] Brown, R. and Loday, J.L., Van Kampen theorems for diagrams of spaces, J. Topol. 26(3) (1987), 311-335.
[11] Ellis, G. and Steiner, R., Higher-dimensional crossed modules and the homotopy groups of (n+1)-ads, J. Pure Appl. Algebra 46 (1987), 117-136.
[12] Gilbert, N.D., Derivations, automorphisms and crossed modules, Comm. Algebra 18(8) (1990), 2703-2734.
[13] Guin-Walery, D. and Loday, J.-L., Obstruction a l'excision en K-theorie algebrique, In: Algebraic K-theory, Lecture Notes in Math. 854, Springer, 1981.
[14] Huebschmann, J., Crossed n-fold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), 302-314.
[15] Icen, I., The equivalence of 2-groupoids and crossed modules, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 49 (2000), 39-48.
[16] Loday, J.-L., Cohomologie et groupe de Steinberg relatifs, J. Algebra 54 (1978), 178- 202.
[17] Loday, J.-L., Spaces with afnitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24(2) (1982), 179-202.
[18] Mackenzie, K., "Lie Groupoids and Lie Algebroids in Differential Geometry", Cam- bridge University Press, 1987.
[19] Maclane, S., "Categories for the Working Mathematician", Springer, 1971.
[20] Mucuk, O. and Demir, S., Normality and quotient in crossed modules over groupoids and double groupoids, Turkish J. Math. 42 (2018), 2336-2347.
[21] Mucuk, O. and Sahan, T., Group-groupoid actions and liftings of crossed modules, Georgian Math. J. 26(3) (2019), 437-447.
[22] Mucuk, O. and Sahan, T., Alemdar, N., Normality and quotients in crossed modules and group-groupoids, Appl. Categ. Structures 23(3) (2015), 415-428.
[23] Norrie, K., Actions and automorphisms of crossed modules, Bull. Soc. Math. France 118 (1990), 129-146.
[24] Temel, S., Normality and quotient in crossed modules over groupoids and 2- groupoids, Korean J. Math. 27(1) (2019), 151-163.
[25] Whitehead, J.H.C., Combinatorial homotopy II, Bull. Amer. Math. Soc. (N.S.) 55 (1949), 453-496.
[26] Whitehead, J.H.C., Note on a previous paper entitled "On adding relations to ho- motopy groups", Ann. of Math. (2) 47 (1946), 806-810.