Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.815719
Title: Higher dimensional algebroids and crossed complexes
Author: Mosa, Ghafar Huseen
Awarding Body: University of Wales, Bangor
Current Institution: Bangor University
Date of Award: 1986
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Abstract:
The equivalence between the category of crossed modules (over groups) and the category of special double groupoids with connections and with one vertix was proved by R.Brown and C.B.Spencer. Also, C. B. Spencer and Y. L. Wong have shown that there exists an equivalence between the category of 2-categories and the category of double categories with connections • R.Brown and P.J.Higgins have generalised the first result: they proved that there exists an equivalence between the category of w-groupoids and the category of crossed complexes (over groupoids). In this thesis we develop a parallel theory in a more algebraic context with expectation of applications in non-abelian homological and homotopical algebra. We prove an equivalence between the category of crossed modules (over algebroids) and the category of special double algebroids with connections' Moreover we prove a similar result for the 3-dimensional case , that is , we prove that there exists an equivalence between the category (Crs) 3 of 3-truncated crossed complexes and the category (w-Alg) 3 of 3-tuple algebroids. Also we end this work by giving a conjecture for the higher dimensional case. In particular, we have Theorem: The functors y, ~ form an adjoint equivalence y: DA! ~-- ➔ C ~ where DA! is the category of special double algebroids with connections and C is the category of crossed modules over algebroids. Theorem: The functors y, >- form an adjoint equivalence y : ( w-A 1 g) n +-- - ➔ ( Cr s ) n : >. for n = 3 , 4. Finally we give a conjecture whose validity would be sufficient for the general equivalence of categories of W-algebroids and crossed complexes. In chapter VI we explain some results which have been obtained in the case of groupoids and higher dimensional groupoids. and suggest the possibility of obtaining similar results in the case of algebroids and higher dimensional algebroids.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.815719  DOI: Not available
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