Action of Crossed Modules and Bar Construction

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.

Keywords

• Bar construction
• (Bi)implicial sets
• Crossed modules
• Moore complex.

References

1. Z. ARVASI and T. PORTER, Freeness conditions for 2-crossed module of commutative algebras, Applied Categorical Structures, 6, (1998), 455-477.
2. P. CARRASCO and A.M. CEGARRA, Group-theoretic algebraic models for homotopy types, Journal of Pure and Applied Algebra, 75, (1991), 195-235.
3. D. CONDUCHÉ, Modules croisés généralisés de longueur 2. Jour. Pure and Applied Algebra, 34, (1984), 155-178.
4. J. DUSKIN, Simplicials methods and the interpretation of triple cohomology, Memoirs A.M.S., Vol.3, (1975), 163.
5. G.J. ELLIS, Higher dimensional crossed modules of algebras, Journal of Pure and Applied Algebra, 52, (1988), 277-282.
6. E. D. FARJOUN and Y. SEGEV, Crossed modules as homotopy normal maps, Topology and its Applications, 157, (2010), 359-368.
7. T. PORTER, Homology of commutative algebras and an invariant of Simis and Vasconceles, Journal of Algebra , 99, (1986), 458-465.
8. 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).

9. 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
10. L.ILLUSIE, Complex cotangent et deformations I, II. Springer Lecture Notes in Math., 239 (1971), II: 283, (1972).
11. M. PREZMA, Homotopy normal maps, http://arxiv.org/pdf/1011.4708v7.pdf, 2012.
12. 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).
13. J.H.C. WHITEHEAD, Combinatorial homotopy , Bull. Amer. Math. Soc., 55, (1949), 453-496.