Comments by Ronnie Brown:
Ghafar Mosa's thesis strikes new ground in dealing with double algebroids and extending the reults of Brown and Spencer on double groupoids to the algebroid (rings with several objects) case.
This extension is not straightforward, as it is not so clear how to axiomatise the connections, to take account of the additive structures and the scalar multiplications. I feel now that a reading of this thesis suggests that what is axiomatised for those structures is the folding operation defined by the connections.
It seemed to me that the availablitity of this thesis has become a necessity, but I have been unable to contact Dr Mosa for his permission. Now that I have the home facilities to do this job, I have taken it upon myself to make the thesis available as a set of pdf files, see below.
The higher dimensional axiomatisation of ω-algebroids was also taken account of in the thesis
F. Al-Agl, Aspects of multiple categories, (1990).
which confined itself to the category rather than algebroid case and itself formed a basis for the paper
R. Brown, F. A. Al-Agl and R. Steiner, ``Multiple categories: the equivalence of a globular and a cubical approach'', Advances in Mathematics 170 (2002) 71-118. pdf
An integration of the two approaches has not yet been done, and may require a lot of work.
An aim of this programme was to go further and, following the model for crossed complexes of groupoids, to develop a monoidal closed structure on ω-algebroids and hence for crossed complexes of algebroids.
|CHAPTER I : R-ALGEBROIDS|
|$2. CROSSED MODULES (OVER ASSOCIATIVE ALGEBRAS)||8|
|$3. CROSSED MODULES (OVER ALGEBROIDS)||10|
|CHAPTER II : DOUBLE R-ALGEBROIDS|
|$2. FUNCTORS (DOUBLE ALGEBROIDS)→(CROSSED MODULES)||23|
|CHAPTER III : THE EQUIVALENCE BETWEEN THE CATEGORY OF CROSSED MODULES
AND THE CATEGORY OF SPECIAL DOUBLE ALGEBROIDS WITH CONNECTIONS
|$1. THIN STRUCTURES AND CONNECTIONS||31|
|$2. THE FOLDING OPERATION||41|
|$3. THE FUNCTOR (CROSSED MODULES) → (DOUBLE ALGEBROIDS)||46|
|$4. THE EQUIVALENCE OF CATEGORIES||52|
|CHAPTER IV : ω-ALGEBROIDS (WITHOUT CONNECTIONS) AND CROSSED COMPLEXES|
|$1. ω-ALGEBROIDS (WITHOUT CONNECTIONS)||60|
|$2. CROSSED COMPLEXES||63|
|$3. THE FUNCTOR (ω-Alg) → (Crs)||66|
|CHAPTER V : THE EQUIVALENCE BETWEEN n-TUPLE ALGEBROIDS
AND CROSSED COMPLEXES FOR n=3 and 4
|$1. ω-ALGEBROIDS WITH CONNECTIONS||72|
|$2. FOLDING OPERATIONS||75|
|$3. COSKELETON OF ω-ALGEBROIDS||98|
|$4. THE EQUIVALENCE OF CATEGORIES||102|
|CHAPTER VI : CONJECTURED RESULTS ON ALGEBROIDS, ω-ALGEBROIDS
AND CROSSED COMPLEXES
|APPENDIX I Verification of Theorem 3.1. and Lemma 3.1.8||108|
|APPENDIX II Verification of proposition 3.2.2 diagrammatically||112|
|APPENDIX III The proof of claim 5.2.16||118|
|APPENDIX IV The proof of proposition 5.2.19||120|
|APPENDIX V The proof of proposition 5.2.20||125|
|APPENDIX VI The proof of proposiion 5.3.2||128|
Downloads as pdf files
Summary and Introduction 2.3 MB
Chapters I - II 3.9 MB
Chapter III and Appendices I-II 4.9 MB
Chapter IV-VI 5.4 MB
References 0.6 MB
1. The purely categorical part of the first three chapters has been extracted and published as
R. Brown and G.H. Mosa `Double categories, 2-categories, thin structures and connections', Theory and Applications of Categories, (1999), 163-175.
2. An abbreviated account of Chapters I-III has been available for some time as a preprint.
May 10, 2009