Higher dimensional algebras as ideal maps

Year 2020, Volume: 49 Issue: 6, 1865 - 1884, 08.12.2020

Alper Odabas Erdal Ulualan

https://doi.org/10.15672/hujms.575080

Abstract

In this work, we explain the close relationship between an ideal map structure $S\rightarrow End_{R}(R)$ on a homomorphism of commutative $k$-algebras $R\rightarrow S$ and an ideal simplicial algebra structure on the associated bar construction $Bar(S,R)$. We also explain this structure for crossed squares of algebras.

Keywords

crossed module, crossed square, ideal map

References

• [1] M. André, Homologie des algèbres commutatives, Die Grundlehren der Mathematischen Wissenchaften, 206 Springer-Verlag 1974.
• [2] Z. Arvasi and T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ., 3, No. 1, pp 1-23, 1997.
• [3] Z. Arvasi and T. Porter, Freeness conditions for 2-crossed module of commutative algebras, Appl. Categ. Structures, 6, 455-471, 1998.
• [4] Z. Arvasi and A. Odabaş, Computing 2-dimensional algebras: Crossed modules and Cat1-algebras, J. Algebra Appl. 15 (10), 1650185, 2016.
• [5] T. Datuashvili and T. Pirashvili, On Co Homology of 2-Types and Crossed Modules, J. Algebra 244, 352-365, 2001.
• [6] G. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra 52, 277-282, 1988.
• [7] G. Ellis, Crossed squares and combinatorial homotopy, Math. Z. 214, 93-110, 1993.
• [8] E.D. Farjoun and K. Hess, Normal and co-normal maps in homotopy theory, Homology Homotopy Appl. 14, 1, 79-112, 2012.
• [9] E.D. Farjoun and Y. Segev, Crossed modules as homotopy normal maps, Topology Appl. 157, 359-368, 2010.
• [10] E.D. Farjoun and Y. Segev, Normal closure and injective normalizer of a group homomorphism, J. Algebra 423, 1010-1043, 2015.
• [11] D. Guin-Waléry and J-L. Loday, Obsructioná l’excision en K-theories algébrique, In: Friedlander, E.M.,Stein, M.R.(eds.) Evanston conf. on algebraic K-Theory 1980, (Lecture Notes in Math., 854, 179-216), Springer, Berlin Heidelberg, 1981.
• [12] L. Illusie, Complex cotangent et deformations I, II, Lecture Notes in Math. 239 1971, II: 283, Springer, 1972.
• [13] T. Porter, Homology of commutative algebras and an invariant of Simis and Vasconceles, J. Algebra 99, 458-465, 1986.
• [14] M. Prezma, Homotopy normal maps, Algebr. Geom. Topol. 12, 1211-1238, 2012.
• [15] N.M. Shammu, Algebraic and categorical structure of category of crossed modules of algebras, University of Wales, PhD Thesis, 1992.
• [16] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949.