TY  - JOUR
T1  - Higher dimensional algebras as ideal maps
AU  - Odabas, Alper
AU  - Ulualan, Erdal
PY  - 2020
DA  - December
DO  - 10.15672/hujms.575080
JF  - Hacettepe Journal of Mathematics and Statistics
PB  - Hacettepe University
WT  - DergiPark
SN  - 2651-477X
SP  - 1865
EP  - 1884
VL  - 49
IS  - 6
LA  - en
AB  - 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.
KW  - crossed module
KW  - crossed square
KW  - ideal map
CR  - [1] M. André, Homologie des algèbres commutatives, Die Grundlehren der Mathematischen
Wissenchaften, 206 Springer-Verlag 1974.
CR  - [2] Z. Arvasi and T. Porter, Higher dimensional Peiffer elements in simplicial commutative
algebras, Theory Appl. Categ., 3, No. 1, pp 1-23, 1997.
CR  - [3] Z. Arvasi and T. Porter, Freeness conditions for 2-crossed module of commutative
algebras, Appl. Categ. Structures, 6, 455-471, 1998.
CR  - [4] Z. Arvasi and A. Odabaş, Computing 2-dimensional algebras: Crossed modules and
Cat1-algebras, J. Algebra Appl. 15 (10), 1650185, 2016.
CR  - [5] T. Datuashvili and T. Pirashvili, On Co Homology of 2-Types and Crossed Modules,
J. Algebra 244, 352-365, 2001.
CR  - [6] G. Ellis, Higher dimensional crossed modules of algebras, J. Pure Appl. Algebra 52,
277-282, 1988.
CR  - [7] G. Ellis, Crossed squares and combinatorial homotopy, Math. Z. 214, 93-110, 1993.
CR  - [8] E.D. Farjoun and K. Hess, Normal and co-normal maps in homotopy theory, Homology
Homotopy Appl. 14, 1, 79-112, 2012.
CR  - [9] E.D. Farjoun and Y. Segev, Crossed modules as homotopy normal maps, Topology
Appl. 157, 359-368, 2010.
CR  - [10] E.D. Farjoun and Y. Segev, Normal closure and injective normalizer of a group homomorphism,
J. Algebra 423, 1010-1043, 2015.
CR  - [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.
CR  - [12] L. Illusie, Complex cotangent et deformations I, II, Lecture Notes in Math. 239 1971,
II: 283, Springer, 1972.
CR  - [13] T. Porter, Homology of commutative algebras and an invariant of Simis and Vasconceles,
J. Algebra 99, 458-465, 1986.
CR  - [14] M. Prezma, Homotopy normal maps, Algebr. Geom. Topol. 12, 1211-1238, 2012.
CR  - [15] N.M. Shammu, Algebraic and categorical structure of category of crossed modules of
algebras, University of Wales, PhD Thesis, 1992.
CR  - [16] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496,
1949.
UR  - https://doi.org/10.15672/hujms.575080
L1  - https://dergipark.org.tr/en/download/article-file/1089447
ER  -