Crossed Corner and Reduced Simplicial Commutative Algebras

Abstract

In this paper, we describe the crossed corner of commutative algebras and present the relation between the category of crossed corners of commutative algebras and the category of reduced simplicial commutative algebras with Moore complex of length 2. We provide a passage from crossed corners to bisimplicial algebras. In this construction, we utilize the Artin-Mazur codiagonal functor from reduced bisimplicial algebras to simplicial algebras and the hypercrossed complex pairings in the Moore complex of a simplicial algebra. Using the coskeleton functor from the category of $k$-truncated simplicial algebras to the category simplicial algebras with Moore complex of length $k$, we see that the length of Moore complex of the reduced simplicial algebra obtained from a crossed corner is 2.

Keywords

• Crossed modules
• simplicial algebras
• crossed corner

References

1. J. H. C. Whitehead, Combinatorial Homotopy II, Bulletin of the American Mathematical Society 55 (1949) 453{--}496.
2. D. Guin-Wal\'{e}ry, J-L. Loday, Obsruction {\'{a}} l'excision en K-theories Alg{\'{e}}brique, in: E. M. Friedlander, M. R. Stein (Eds.), Algebraic K-Theory Evanston 1980, Vol. 854 of \emph{Lectute Notes Mathematics}, Springer, Berlin, 1981, pp. 179{--}216.
3. H. J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, Walter de Gruyter, Berlin, 1991.
4. Z. Arvasi, E. Ulualan, Quadratic and 2-Crossed Modules of Algebras, Algebra Colloquium 14 (2007) 669{--}686.
5. E. Ulualan, E. Uslu, Quadratic Modules for Lie Algebras, Hacettepe Journal of Mathematics and Statistics 40 (3) (2011) 409{--}419.
6. E. Özel, U. E. Arslan, On Quasi Quadratic Modules of Lie Algebras, Journal of New Theory (41) (2022) 62{--}69.
7. E. Soylu Yılmaz, K. Yılmaz, On Relations among Quadratic Modules, Mathematical Methods in the Applied Sciences 45 (18) (2022) 12231{--}12244.
8. M. Alp, Characterization of Crossed Corner, Algebras, Groups and Geometries 16 (2) (1999) 173{--}182.

9. T. Porter, Homology of Commutative Algebras and an Invariant of Simis and Vasconceles, Journal of Algebra 99 (1986) 458{--}465.
10. G. J. Ellis, Higher Dimensional Crossed Modules of Algebras, Journal of Pure and Applied Algebra {52} (1988) 277{--}282.
11. Z. Arvasi, Crossed Squares and 2-Crossed Modules of Commutative Algebras, Theory and Applications of Categories 3 (7) (1997) 160{--}181.
12. Z. Arvasi, T. Porter, Higher Dimensional Peiffer Elements in Simplicial Commutative Algebras, Theory and Applications of Categories 3 (1) (1997) 1{--}23.
13. N. M. Shammu, Algebraic and Categorical Structure of Categories of Crossed Modules of Algebras, Doctoral Dissertation North Carolina Wilmington University (1992) Bangor.
14. I. Akça, Z. Arvasi, Simplicial and Crossed Lie Algebras, Homology, Homotopy and Applications 4 (1) (2002) 43{--}57.
15. A. Aytekin, Categorical Structures of Lie-Rinehart Crossed Module, Turkish Journal of Mathematics 43 (1) (2019) 511{--}522.
16. M. Alp, Applications of Crossed Corner, Algebras, Groups and Geometries 16 (2) (1999) 337{--}344.
17. M. Alp, A. Bekir, E. Ulualan, Relation Between Crossed Square and Crossed Corner, Journal of Science and Technology of Dumlupınar University (002) (2001) 89{--}96.
18. Ö. Gürmen Alansal, E. Ulualan, Bisimplicial Commutative Algebras and Crossed Squares, Fundamental Journal of Mathematics and Applications 6 (2023) 177{--}187.
19. M. Artin, B. Mazur, On the Van Kampen Theorem, Topology 5 (1966) 179{--}189.