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# XModAlg

## Crossed Modules and Cat1-Algebras

1.18

17 November 2020

Zekeriya Arvasi
Email: zarvasi@ogu.edu.tr
Address:
Prof. Dr. Z. Arvasi
Osmangazi University
Arts and Sciences Faculty
Department of Mathematics and Computer Science
Eskisehir
Turkey

Alper Odabas
Email: aodabas@ogu.edu.tr
Address:
Dr. A. Odabas
Osmangazi University
Arts and Sciences Faculty
Department of Mathematics and Computer Science
Eskisehir
Turkey

### Abstract

The XModAlg package provides functions for computation with crossed modules of commutative algebras and cat\(^{1}\)-algebras.

Bug reports, suggestions and comments are, of course, welcome. Please submit an issue on GitHub at http://github.com/gap-packages/xmodalg/issues/ or contact the second author at aodabas@ogu.edu.tr.

### Copyright

© 2014-2020, Zekeriya Arvasi and Alper Odabas.

The XModAlg package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

### Acknowledgements

This documentation was prepared with the GAPDoc [LN17] and AutoDoc [GH17] packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor17] and the package ReleaseTools.

Both authors are very grateful to Chris Wensley (http://pages.bangor.ac.uk/~mas023/) for helpful suggestions.

This work was partially supported by TÜBİTAK (The Scientific and Technical Research Council of Turkey), project number 107T542.

### Contents

1 Introduction
2 Cat1-algebras
2.1 Definitions and examples
2.1-1 Cat1Algebra
2.1-2 Source
2.1-3 Cat1AlgebraSelect
2.1-4 SubCat1Algebra
2.2 Cat\(^{1}-\)algebra morphisms
2.2-1 Cat1AlgebraMorphism
2.2-2 Source
3 Crossed modules
3.1 Definition and Examples
3.1-1 XModAlgebra
3.1-2 Source
3.1-3 SubXModAlgebra
3.1-4 PreXModAlgebraByBoundaryAndAction
3.2 (Pre-)Crossed Module Morphisms
3.2-1 XModAlgebraMorphism
3.2-2 Kernel
3.2-3 Image
3.2-4 SourceHom
4 Conversion between cat1-algebras and crossed modules
4.1 Equivalent Categories
4.1-1 Cat1AlgebraOfXModAlgebra
4.1-2 XModAlgebraOfCat1Algebra
References
Index
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