New Perspectives in Algebra, Topology and Categories: Summer School, Louvain-la-Neuve, Belgium, September 12-15, 2018 and September 11-14, 2019

Editor’s Preface
Preface to the Book Series Coimbra Mathematical Texts
Contents
1 Ring Epimorphisms, Gabriel Topologies and Contramodules
	1 Preliminaries
	2 Ring Epimorphisms
	3 Gabriel Topologies, Torsion Pairs and the Ring of Quotients
	4 Comodules and Contramodules
		4.1 Coalgebras, Comodules, Contramodules
	5 First Matlis Category Equivalence
	References
2 An Invitation to  Topological Semi-abelian Algebras
	1 Semi-abelian Algebras
		1.1 Semi-abelian Theories
		1.2 Semi-abelian Algebras: Examples
		1.3 Semi-abelian Algebras: Some Properties
		1.4 Topological Algebras
	2 Topological Semi-abelian Algebras
		2.1 How to Overcome Lack of Homogeneity
		2.2 The Closure on Subalgebras
		2.3 Quotient Maps
		2.4 Separation Properties
		2.5 (Local) Compactness
		2.6 Connectedness and Total Disconnectedness
	3 The Categorical Behaviour of Topological Semi-abelian Algebras
		3.1 Properties of the Category TopmathbbT
		3.2 Special Subcategories of TopmathbbT
	4 Split Extensions: Semi-direct Products
		4.1 Semidirect Products of Groups
		4.2 Semidirect Products of Semi-abelian Algebras
		4.3 Semidirect Products of Topological Semi-abelian Algebras
	5 Split Extensions: Classifiers
		5.1 Groups have Split Extension Classifiers
		5.2 A Digression through Split Extension Classifiers for Internal Groups
		5.3 Topological Groups have Split Extensions Classifiers
	6 Some Open Problems
		6.1 Coproducts of Topological Algebras
		6.2 Split Extension Classifiers of Topological Algebras
		6.3 Split Extension Classifiers: Topological Lie Algebras
		6.4 Algebraic Coherence for Topological Groups
		6.5 Local Algebraic Cartesian Closedness for Topological Groups
	References
3 Commutative Monoids, Noncommutative Rings and Modules
	1 Commutative Monoids
		1.1 Commutative Monoids and Their Morphisms
		1.2 Preorders
		1.3 Congruences
		1.4 The Additive Monoid mathbbN0 of Natural Numbers
		1.5 Congruences in the Monoid mathbbN0
		1.6 Prime Ideals and Localizations
	2 Preordered Groups, Positive Cones
	3 Some Set Theory
		3.1 ZFC
		3.2 Grothendieck's Universes
		3.3 NBG
	4 The Monoid V(mathscrC), Discrete Valuations, Krull Monoids
		4.1 The Monoid V(mathscrC)
		4.2 Discrete Valuations, Krull Monoids
	5 Modules
		5.1 Left Modules
		5.2 Right Modules
		5.3 Abelian Groups = mathbbZ-modules
		5.4 Is Left Better Than Right?
		5.5 Two Exercises
	6 Representations/Modules/Actions of Other Algebraic Structures
		6.1 k-algebras
		6.2 Lie k-algebras
		6.3 Monoids
		6.4 Monoids with Zero
		6.5 Near-Rings
		6.6 Groups and the Cayley Representation
		6.7 Groups G and Action of G on G via Inner Automorphisms
	7 Free Modules
		7.1 Definition and First Properties of Free Modules
		7.2 Crash Course of Linear Algebra over Non-commutative Division Rings
		7.3 Rank of a Free Module
	8 IBN Rings
	9 Simple Modules, Semisimple Modules
	10 Projective Modules
		10.1 The Ring of ntimesn Matrices over a Division Ring
	11 Superfluous Submodules and Radical of a Module
	12 The Jacobson Radical of a Ring
	13 Injective Modules
	14 Projective Covers
	15 Injective Envelopes
	16 The Monoid V(R)
	References
4 An Introduction to Regular Categories
	1 Regular Categories
		1.1 Strong and Regular Epimorphisms
		1.2 Quotients in Algebraic Categories
		1.3 Examples of Regular Categories
		1.4 Canonical Factorization
		1.5 The Barr-Kock Theorem
	2 Relations in Regular Categories
	3 Calculus of Relations and Mal'tsev Categories
		3.1 Examples of Mal'tsev Categories
		3.2 Regular Pushouts
	4 Goursat Categories
		4.1 Goursat Pushouts
		4.2 Implication Algebras
		4.3 Diagram Lemmas and Goursat Categories
	References
5 Categorical Commutator Theory
	1 Commutators of Groups
	2 The Case of Ω-groups
	3 The Categorical Higgins Commutator
	4 The Huq Commutator
	5 Abelian Objects
	References
6 Notes on Point-Free Topology
	1 Prologue
		1.1 General Topology
		1.2 A Synthetic Generalized Geometry
		1.3 A More Realistic Account of the Events
		1.4 Frame Homomorphisms
	2 Background
		2.1 Posets
		2.2 Adjunctions
		2.3 For Category Minded Readers: Posets as Special Categories
		2.4 Some Special Posets
		2.5 Pseudocomplements, Supplements and Complements
		2.6 Heyting Algebras
		2.7 Boolean Algebras
	3 Frames and Spaces
		3.1 The Category of Frames
		3.2 Spaces and Frames. The Functor Ω
		3.3 The Heyting Structure
		3.4 Prime Elements and Sobriety
		3.5 Theorem
		3.6 Points and Spectra
		3.7 Spatial Frames
		3.8 Sober Reflection
		3.9 Classical and Generalized (Point-Free) Spaces
	4 Categorical Remarks
		4.1 Semilattices and a Free Functor
		4.2 Free Objects in Frm
		4.3 Algebraic Aspects of Frm
		4.4 Taking Quotients
		4.5 Product in Loc (Coproduct in Frm) Concretely
	5 Loc as a Concrete Category. Localic Maps and Sublocales
		5.1 Localic maps
		5.2 Proposition
		5.3 Aside: Spectrum in Thus Represented Category of Locales
		5.4 Sublocales and the Coframe S(L)
		5.5 Open and Closed Sublocales
		5.6 Closure, Density and Isbell's Theorem. Interior
		5.7 Subspaces and Sublocales I. The Axiom TD
		5.8 Aside: Spatialization as a Sublocale
	6 Images and Preimages. Localic Maps as Continuous Ones. Open Maps
		6.1 Proposition
		6.2 Localic Preimage
		6.3 Proposition
		6.4 Points, Sublocales and Subspaces
		6.5 Geometry of Localic Maps
		6.6 Joyal-Tierney Theorem
	7 Examples
		7.1 Regularity
		7.2 Dense Maps
		7.3 Theorem
		7.4 Complete Regularity
		7.5 A Point-Free Stone-Čech Compactification
		7.6 A Glimpse of Other Separation Axioms
		7.7 A Few More Examples
	References
7 Non-associative Algebras
	1 Non-associative Algebras
	2 Examples
	3 The Zero Algebra; Kernels and Cokernels
	4 Kernels and Ideals, Cokernels and Quotients
	5 Short Exact Sequences and Protomodularity
	6 Polynomials and Free Non-associative Algebras
	7 Varieties of Non-associative Algebras
	8 Regularity, Exact Sequences
	9 Semi-abelian Categories
	10 Birkhoff Subcategories
	11 Homogeneous Identities
	12 Some Recent Results
	13 Bibliography
	References