Korean J. Math.  Vol 26, No 3 (2018)  pp.439-458
DOI: https://doi.org/10.11568/kjm.2018.26.3.439

Derived crossed modules

Tunçar Şahan

Abstract


In this study, we interpret the notion of homotopy of morphisms in the category of crossed modules in a category $\mathsf{C}$ of groups with operations using the categorical equivalence between the categories of crossed modules and of internal categories in $\mathsf{C}$. Further, we characterize the derivations of crossed modules in a category $\mathsf{C}$ and obtain new crossed modules using regular derivations of old one.

Keywords


Homotopy, crossed module, internal category, group with operations.

Subject classification

55U35, 18D35, 20L05, 18A23.

Sponsor(s)



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References


H. F. Akız, O. Mucuk, N. Alemdar and T. S ̧ahan, Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J. 20 (2) (2013), 223–238. (Google Scholar)

J. C. Baez and D. Stevenson, The Classifying Space of a Topological 2-Group, pages 1–31. Algebraic Topology. Abel Symposia. Springer, 2009. (Google Scholar)

R. Brown, Groupoids and crossed objects in algebraic topology Homol. Homotopy Appl. 1 (1999), 1–78. (Google Scholar)

R. Brown and J. Huebschmann, Identities among relations, pages 153–202. Lon- don Mathematical Society Lecture Note Series. Cambridge University Press, 1982. (Google Scholar)

R. Brown and C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indagat. Math. 79 (4) (1976), 296–302. (Google Scholar)

W. H. Cockcroft, On the homomorphisms of sequences, Math. Proc. Cambridge 48 (4) (1952), 521–532. (Google Scholar)

T. Datuashvili, Cohomologically trivial internal categories in categories of groups with operations, Appl. Categor. Struct. 3 (3) (1995), 221–237. (Google Scholar)

T. Datuashvili. Categorical, homological and homotopical properties of algebraic objects. Dissertation, Georgian Academy of Science, 2006. (Google Scholar)

J. Huebschmann, Crossed n-folds extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), 302–313. (Google Scholar)

J.-L. Loday, Cohomologie et groupe de steinberg relatifs, J. Algebra 54 (1) (1978), 178–202. (Google Scholar)

A. S.-T. Lue, Cohomology of groups relative to a variety, J. Algebra 69 (1) (1981), 155–174. (Google Scholar)

O. Mucuk and H.F. Akız, Monodromy groupoid of an internal groupoid in topological groups with operations, Filomat 29 (10), (2015), 2355–2366. (Google Scholar)

O. Mucuk and H. C ̧ akallı, G-connectedness in topological groups with operations, 1079-1089 Filomat 32 (3), (2018), 1079–1089. (Google Scholar)

O. Mucuk and T. S ̧ahan, Coverings and crossed modules of topological groups with operations, Turk. J. Math. 38 (5) (2014), 833–845. (Google Scholar)

K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. Fr. 118 (2) (1990), 129–146. (Google Scholar)

G. Orzech, Obstruction theory in algebraic categories, I, J. Pure. Appl. Algebra 2 (4) (1972), 287–314. (Google Scholar)

G. Orzech, Obstruction theory in algebraic categories, II, J. Pure. Appl. Algebra 2 (4) (1972), 315–340. (Google Scholar)

A. Patchkoria, Crossed semimodules and schreier internal categories in the cat- egory of monoids, Georgian Math. J. 5 (6) (1998), 575–581. (Google Scholar)

T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations P. Edinburgh. Math. Soc. 30 (3) (1987), 373–381. (Google Scholar)

T. S ̧ahan, Further remarks on liftings of crossed modules, Hacet. J. Math. Stat., Retrieved 4 Mar. 2018, from Doi: 10.15672/HJMS.2018.554. (Google Scholar)

S. Temel, Topological crossed semimodules and schreier internal categories in the category of topological monoids, Gazi Univ. J. Sci. 29 (4) (2016), 915–921. (Google Scholar)

S. Temel, Crossed semimodules of categories and Schreier 2-categories, Tbilisi Math. J. 11 (2) (2018), 47–57. (Google Scholar)

J. H. C. Whitehead, Note on a previous paper entitled ”on adding relations to homotopy groups”, Ann. Math. 47 (4) (1946), 806–810. (Google Scholar)

J. H. C. Whitehead, On operators in relative homotopy groups, Ann. Math. 49 (3) (1948), 610–640. (Google Scholar)

J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (5) (1949), 453–496. (Google Scholar)


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