Bisimplicial Commutative Algebras and Crossed Squares

Year 2023, Volume: 6 Issue: 3, 177 - 187, 30.09.2023

Özgün Gürmen Alansal Erdal Ulualan

https://doi.org/10.33401/fujma.1292885

Cited By: 2

Abstract

A simplicial commutative algebra with Moore complex of length 1 gives a crossed module structure over commutative algebras. In this study, we will give 2-dimensional version of this result by giving hypercrossed complex pairings for a bisimplicial algebra and its Moore bicomplex. We give a detailed calculation in low dimensions for these pairings to see their role in the structures of crossed squares and bisimplicial algebras. In this context, we prove that if the Moore bicomplex of a bisimplicial commutative algebra is of length 1, then it gives a crossed square structure over commutative algebras.

Keywords

Bisimplicial algebras, Crossed modules, Moore complex

References

• [1] A.R. Grandjean, M.J. Vale, 2-Modulos cruzados an la cohomologia de andr´e-quillen, Memorias de la Real Academia de Ciencias, 22 (1986), 1–28.
• [2] D. Conduche, Modules crois´es c´en´eralis´es de longueur 2, J. Pure Appl. Algebra, 34 (1984), 155-178.
• [3] D. Conduche, Simplicial crossed modules and mapping cones, Georgian Math. J., 10(4) (2003), 623-636.
• [4] D. Guin-Wal´ery, J-L. Loday, Obsruction´a l’excision en K-theories Alg´ebrique, Evanston conference on Algebraic K-Theory 1980, (Lectute Notes Mathematics) 854, 179-216, Berlin Heidelberg New York, Springer 1981.
• [5] G.J. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra, 52 (1988), 277-282.
• [6] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453-496.
• [7] L. Illusie, Complex Cotangent et Deformations I, II, Springer Lecture Notes in Mathematics 239 1971 II 283 1972.
• [8] M. Andre, Homologie des Algebres Commutatives, Die Grundlehren der Mathematischen Wissenchaften, 206 New York, Springer-Verlag, 1974.
• [9] Ö. Gürmen Alansal, E. Ulualan, Peiffer pairings in multisimplicial groups and crossed n-cubes and applications for bisimplicial groups, Turkish J. Math., 45(1) (2021), 360-386.
• [10] P. Carrasco, A.M. Cegarra, Group-theoretic algebraic models for homotopy types, J. Pure Appl. Algebra, 75 (1991), 19-235.
• [11] T. Porter, Homology of commutative algebras and an invariant of simis and vasconceles, J. Algebr., 99 (1986), 458-465.
• [12] T. Porter, n-type of simplicial groups and crossed n-cubes, Topol., 32 (1993), 5-24.
• [13] Z. Arvasi, T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ., 3(1) (1997), 1-23.
• [14] Z. Arvasi, Crossed squares and 2-crossed modules of commutative algebras, Theory Appl. Categ., 3(7) (1997), 160-181.