Research Article
Year 2025,
Volume: 7 Issue: 1, 49 - 61, 24.07.2025
Abstract
If a group $N$ acts on a set $X$, a simplicial set $Bar(X,N)$ using the usual bar construction has been provided. In this construction, if the group $N$ acts on a group $G$ via a homomorphism $f:N\rightarrow G$, then $Bar(G,N)$ has a simplicial set structure. In the case of $f$ has a crossed module structure, $Bar(G,N)$ has a normal simplicial group structure. In this work, by defining an action of a crossed module $\partial: N_1 \longrightarrow X_1$ on a homomorphism of groups $f: N_2 \longrightarrow X_2 $ via a double map $\alpha: \partial\rightarrow f$, we will construct a bisimplicial set, using the 2-dimensional version of the usual Bar construction.
References
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Year 2025,
Volume: 7 Issue: 1, 49 - 61, 24.07.2025
Abstract
References
- Z. ARVASI and T. PORTER, Freeness conditions for 2-crossed module of commutative algebras, Applied Categorical Structures, 6, (1998), 455-477.
- P. CARRASCO and A.M. CEGARRA, Group-theoretic algebraic models for homotopy types, Journal of Pure and Applied Algebra, 75, (1991), 195-235.
- D. CONDUCHÉ, Modules croisés généralisés de longueur 2. Jour. Pure and Applied Algebra, 34, (1984), 155-178.
- J. DUSKIN, Simplicials methods and the interpretation of triple cohomology, Memoirs A.M.S., Vol.3, (1975), 163.
- G.J. ELLIS, Higher dimensional crossed modules of algebras, Journal of Pure and Applied Algebra, 52, (1988), 277-282.
- E. D. FARJOUN and Y. SEGEV, Crossed modules as homotopy normal maps, Topology and its Applications, 157, (2010), 359-368.
- T. PORTER, Homology of commutative algebras and an invariant of Simis and Vasconceles, Journal of Algebra , 99, (1986), 458-465.
- T. PORTER The Crossed Menagerie: an introduction to crossed gadgetry and co-homology in algebra and topology, (2011), (available from the n-Lab, http://ncatlab.org/nlab/show/Menagerie).
- W. DWYER and E.D. FARJOUN, The localization and cellularization of principal fibrations, Alpine perspectives on algebraic topology, 117-124, Contemp. Math., 504, Amer. Math. Soc., Providence, RI, 2009, available from http://www3.nd.edu/~wgd/drafts/cellular.pdf
- L.ILLUSIE, Complex cotangent et deformations I, II. Springer Lecture Notes in Math., 239 (1971), II: 283, (1972).
- M. PREZMA, Homotopy normal maps, http://arxiv.org/pdf/1011.4708v7.pdf, 2012.
- 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, (Lect. Notes Math., vol.854, pp 179- 216), Berlin Heidelberg New York: Springer (1981).
- J.H.C. WHITEHEAD, Combinatorial homotopy , Bull. Amer. Math. Soc., 55, (1949), 453-496.
There are 13 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory, Category Theory, K Theory, Homological Algebra |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | May 25, 2025 |
| Publication Date | July 24, 2025 |
| Submission Date | August 20, 2024 |
| Acceptance Date | October 30, 2024 |
| Published in Issue | Year 2025 Volume: 7 Issue: 1 |
