Cat$^1$-Objects in the Category of Crossed Modules

Tunçar Şahan; Bestami Berberoğlu

doi:10.47000/tjmcs.1833989

Cat$^1$-Objects in the Category of Crossed Modules

Abstract

The main aim of this study is to investigate cat$^1$-objects within the category of crossed modules, which we call cat$^1$-crossed modules, and to characterize these structures systematically. First, fundamental notions from category theory and crossed modules are presented, followed by the definition and properties of cat$^1$-groups. Subsequently, cat$^1$-crossed modules are introduced in the category of crossed modules, and their homotopical and algebraic properties are explored. Furthermore, natural equivalences are studied in detail between cat$^1$-crossed modules and other algebraic structures such as crossed squares, cat$^2$-groups, and double group-groupoids. The results obtained represent a significant step toward a deeper understanding of higher-dimensional algebraic models.

Keywords

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Details

Primary Language

English

Subjects

Category Theory, K Theory, Homological Algebra

Journal Section

Research Article

Authors

Tunçar Şahan ^*
0000-0002-6552-4695
Türkiye

Bestami Berberoğlu
0000-0002-5140-1600
Türkiye

Publication Date

October 13, 2026

Submission Date

December 1, 2025

Acceptance Date

March 27, 2026

Published in Issue

Year 2026 Volume: 18 Number: 3

DOI

https://doi.org/10.47000/tjmcs.1833989

IZ

https://izlik.org/JA69RE99PW

Cite

RIS / Bibtex

APA

Şahan, T., & Berberoğlu, B. (2026). Cat$^1$-Objects in the Category of Crossed Modules. Turkish Journal of Mathematics and Computer Science, 18(3), 722-734. https://doi.org/10.47000/tjmcs.1833989

AMA

1.Şahan T, Berberoğlu B. Cat$^1$-Objects in the Category of Crossed Modules. TJMCS. 2026;18(3):722-734. doi:10.47000/tjmcs.1833989

Chicago

Şahan, Tunçar, and Bestami Berberoğlu. 2026. “Cat$^1$-Objects in the Category of Crossed Modules”. Turkish Journal of Mathematics and Computer Science 18 (3): 722-34. https://doi.org/10.47000/tjmcs.1833989.

EndNote

Şahan T, Berberoğlu B (October 1, 2026) Cat$^1$-Objects in the Category of Crossed Modules. Turkish Journal of Mathematics and Computer Science 18 3 722–734.

IEEE

[1]T. Şahan and B. Berberoğlu, “Cat$^1$-Objects in the Category of Crossed Modules”, TJMCS, vol. 18, no. 3, pp. 722–734, Oct. 2026, doi: 10.47000/tjmcs.1833989.

ISNAD

Şahan, Tunçar - Berberoğlu, Bestami. “Cat$^1$-Objects in the Category of Crossed Modules”. Turkish Journal of Mathematics and Computer Science 18/3 (October 1, 2026): 722-734. https://doi.org/10.47000/tjmcs.1833989.

JAMA

1.Şahan T, Berberoğlu B. Cat$^1$-Objects in the Category of Crossed Modules. TJMCS. 2026;18:722–734.

MLA

Şahan, Tunçar, and Bestami Berberoğlu. “Cat$^1$-Objects in the Category of Crossed Modules”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 3, Oct. 2026, pp. 722-34, doi:10.47000/tjmcs.1833989.

Vancouver

1.Tunçar Şahan, Bestami Berberoğlu. Cat$^1$-Objects in the Category of Crossed Modules. TJMCS. 2026 Oct. 1;18(3):722-34. doi:10.47000/tjmcs.1833989