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 ωalgebroids was also taken account of in the thesis
F. AlAgl, 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. AlAgl and R. Steiner, ``Multiple categories: the equivalence of a globular and a cubical approach'', Advances in Mathematics 170 (2002) 71118. 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 ωalgebroids and hence for crossed complexes of algebroids.
Thesis contents:
SUMMARY  
CONTENTS  
INTRODUCTION  ixi 
CHAPTER I : RALGEBROIDS  
$0. INTRODUCTION  1 
$2. CROSSED MODULES (OVER ASSOCIATIVE ALGEBRAS)  8 
$3. CROSSED MODULES (OVER ALGEBROIDS)  10 
CHAPTER II : DOUBLE RALGEBROIDS  
$0. INTRODUCTION  15 
15  
$2. FUNCTORS (DOUBLE ALGEBROIDS)→(CROSSED MODULES)  23 
$3. EXAMPLES  26 
CHAPTER III : THE EQUIVALENCE BETWEEN THE CATEGORY OF CROSSED MODULES AND THE CATEGORY OF SPECIAL DOUBLE ALGEBROIDS WITH CONNECTIONS 

$0. INTRODUCTION  30 
$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 
$5. REFLECTION  56 
CHAPTER IV : ωALGEBROIDS (WITHOUT CONNECTIONS) AND CROSSED COMPLEXES  
$0. INTRODUCTION  60 
$1. ωALGEBROIDS (WITHOUT CONNECTIONS)  60 
$2. CROSSED COMPLEXES  63 
$3. THE FUNCTOR (ωAlg) → (Crs)  66 
CHAPTER V : THE EQUIVALENCE BETWEEN nTUPLE ALGEBROIDS AND CROSSED COMPLEXES FOR n=3 and 4 

$0. INTRODUCTION  72 
$1. ωALGEBROIDS WITH CONNECTIONS  72 
$2. FOLDING OPERATIONS  75 
$3. COSKELETON OF ωALGEBROIDS  98 
$4. THE EQUIVALENCE OF CATEGORIES  102 
CHAPTER VI : CONJECTURED RESULTS ON ALGEBROIDS, ωALGEBROIDS AND CROSSED COMPLEXES 
105 
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 
REFERENCES  134 
Downloads as pdf files
Summary and Introduction 2.3 MB
Chapters I  II 3.9 MB
Chapter III and Appendices III 4.9 MB
Chapter IVVI 5.4 MB
References 0.6 MB
Notes:
1. The purely categorical part of the first three chapters has been extracted and published as
R. Brown and G.H. Mosa `Double categories, 2categories, thin structures and connections', Theory and Applications of Categories, (1999), 163175.
2. An abbreviated account of Chapters IIII has been available for some time as a preprint.
May 10, 2009