Internal Categories in Crossed Semimodules and Schreier Internal Categories

Sedat Temel

doi:10.36753/mathenot.691956

EN

Internal Categories in Crossed Semimodules and Schreier Internal Categories

Abstract

In this paper, we characterize internal categories in the category of crossed semimodules and the category of Schreier internal categories within monoids. Then we prove a natural equivalence between their categories. This allows us to produce various examples of double categories. .

Keywords

Crossed module, Crossed semimodule, Schreier internal category, Double category.

References

[1] Baez, J.C., Baratin, A., Freidel, L. and Wise, D.K.: Infinite-Dimensional Representations of 2-Groups. Mem. Am. Math. Soc. 219, (1032) (2012).
[2] Baez, J.C., Lauda, A.D.: Higher Dimensional Algebra V: 2-Groups. Theory Appl. Categ. 12, 423–491 (2004).
[3] Brown, R.: Topology and Groupoids. BookSurge LLC, North Carolina (2006).
[4] Brown, R, Spencer, C.B.: Double groupoids and crossed modules. Cahiers de Topologie et Géométrie Différentielle Catégoriques 17 (4), 343-362 (1976).
[5] Brown, R., Spencer, C.B.: G-groupoids, crossed modules and the fundamental groupoid of a topological group. Mathematical Sciences and Applications E-Notes. Indagat. Math. 79 (4), 296-302 (1976).
[6] 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).
[7] Brown, R., Mucuk, O.: Covering groups of non-connected topological groups revisited. Math. Proc. Camb. Phil. Soc. 115, 97–110 (1994).
[8] Ehresmann, C.: Catégories doubles et catégories structurées. C. R. Acad. Sci. Paris 256, 1198-1201 (1963).
[9] Ehresmann, C.: Catégories structurées. Ann. Sci. Ec. Norm. Super. 80, 349-425 (1963b).
[10] Huebschmann, J.: Crossed n-fold extensions of groups and cohomology. Comment. Math. Helvetici. 55: 302-314 (1980).

[11] Kerler, T. and Lyubashenko, V.V.: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Springer-Verlag. Berlin, Heidelberg, (2001).
[12] Loday, J.-L.: Cohomologie et groupes de Steinberg relatifs. J. Algebra. 54 178-202 (1978).
[13] Maclane, S.: Categories for the Working Mathematician, Graduate Text in Mathematics. 5, Springer-Verlag. New York (1971).
[14] Mucuk, O., Demir S.: Normality and quotient in crossed modules over groupoids and double groupoids. Turk J Math, 42, 2336 – 2347 (2018).
[15] Patchkoria, A.: Crossed Semimodules and Schreier Internal Categories In The Category of Monoids. Georgian Math. J. 5(6), 575-581 (1998).
[16] Porter, T.: Crossed Modules in Cat and a Brown-Spencer Theorem for 2-Categories. Cah. Topol. Géom. Différ. Catég. XXVI-4 (1985).
[17] ¸Sahan, T., Mohammed, J.J.: Categories internal to crossed modules Sakarya University Journal of Science. 23 (4), 519-531, (2019).
[18] Temel, S., ¸Sahan, T. and Mucuk, O.: Crossed modules, double group-groupoids and crossed squares. Preprint arxiv:1802.03978v2 (2018).
[19] 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).
[20] Temel, S.: Crossed semimodules of categories and Schreier 2-categories. Tbilisi Math. J. 11 (2), 47-57 (2018).
[21] Temel, S.: Normality and quotient in crossed modules over groupoids and 2-groupoids. Korean J. Math. 27 (1), 151-163 (2018).
[22] Whitehead, J.H.C.: Combinatorial homotopy II. Bull. Amer. Math. Soc. 55, 453-496 (1949).
[23] Whitehead, J.H.C.: Note on a previous paper entitled "On adding relations to homotopy group". Ann. Math. 47,806-810 (1946).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Sedat Temel ^*
0000-0001-6553-8758
Türkiye

Publication Date

October 15, 2020

Submission Date

February 21, 2020

Acceptance Date

April 17, 2020

Published in Issue

Year 2020 Volume: 8 Number: 2

DOI

https://doi.org/10.36753/mathenot.691956

IZ

https://izlik.org/JA63GT26LR

APA

Temel, S. (2020). Internal Categories in Crossed Semimodules and Schreier Internal Categories. Mathematical Sciences and Applications E-Notes, 8(2), 86-95. https://doi.org/10.36753/mathenot.691956

AMA

1.Temel S. Internal Categories in Crossed Semimodules and Schreier Internal Categories. Math. Sci. Appl. E-Notes. 2020;8(2):86-95. doi:10.36753/mathenot.691956

Chicago

Temel, Sedat. 2020. “Internal Categories in Crossed Semimodules and Schreier Internal Categories”. Mathematical Sciences and Applications E-Notes 8 (2): 86-95. https://doi.org/10.36753/mathenot.691956.

EndNote

Temel S (October 1, 2020) Internal Categories in Crossed Semimodules and Schreier Internal Categories. Mathematical Sciences and Applications E-Notes 8 2 86–95.

IEEE

[1]S. Temel, “Internal Categories in Crossed Semimodules and Schreier Internal Categories”, Math. Sci. Appl. E-Notes, vol. 8, no. 2, pp. 86–95, Oct. 2020, doi: 10.36753/mathenot.691956.

ISNAD

Temel, Sedat. “Internal Categories in Crossed Semimodules and Schreier Internal Categories”. Mathematical Sciences and Applications E-Notes 8/2 (October 1, 2020): 86-95. https://doi.org/10.36753/mathenot.691956.

JAMA

1.Temel S. Internal Categories in Crossed Semimodules and Schreier Internal Categories. Math. Sci. Appl. E-Notes. 2020;8:86–95.

MLA

Temel, Sedat. “Internal Categories in Crossed Semimodules and Schreier Internal Categories”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, Oct. 2020, pp. 86-95, doi:10.36753/mathenot.691956.

Vancouver

1.Sedat Temel. Internal Categories in Crossed Semimodules and Schreier Internal Categories. Math. Sci. Appl. E-Notes. 2020 Oct. 1;8(2):86-95. doi:10.36753/mathenot.691956

Cited By

Further remarks on group-2-groupoids

Applied General Topology

https://doi.org/10.4995/agt.2021.13148

INDEXING & ABSTRACTING & ARCHIVING

20477 The published articles in MSAEN are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.