Category Theory and Applications

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• Categories
• Universal properties
• Diagrams in a category
• Functors
• Natural transformations and adjunctions
• A brief outline
• Classes of categories
• Our approach
• Literature
• Notation
• Acknowledgements

The following sections are included:

• Categories
• Monoids and preordered sets as categories
• Monomorphisms and epimorphisms
• Functors
• Natural transformations and equivalence of categories
• Categories of functors, representable functors
• A category of Galois connections
• Elementary categories of relations and partial mappings

Limits and colimits in a category C are defined by a universal property, which determines a solution up to isomorphism. The general notion of a limit (introduced in Section 2.2) includes cartesian products, equalisers and pullbacks (analysed in Sections 2.1 and 2.3). Dually, colimits comprise sums, coequalisers, pushouts and the classical ‘inductive limits’ (analysed in Sections 2.1, 2.3 and 2.6). The classical ‘projective limits’ are also included in general limits, but – in the usual categories of structured sets – are not worthy of a separate study…

Adjunctions, a crucial step in category theory, were introduced by Kan in 1958 [K2]. They extend Galois connections (defined in Section 1.7) from ordered sets to general categories. The closely related notion of ‘monad’ is also studied here…

We begin by studying, in Section 4.1, the free structures for various categories of algebraic kind. Regular and Barr-exact categories are introduced in Section 4.2; their categories of relations in Section 4.5…

Various categorical topics have interesting applications in Topology and Algebraic Topology. One of the first interactions of this kind can be found in Eilenberg–Steenrod’s foundations of (topological) homology theories [EiS], whose axioms deal with categories of ‘pairs’ of topological spaces, homology functors and natural transformations. (Something about this will be said in 6.1.5.)…

In this chapter we present some hints at a particular approach to Homological Algebra, exposed in the book [G9] and further extended in [G10]. This presentation cannot be self-contained, and various proofs will be referred to [G9]…

We sketch here the main two-dimensional categorical structures, namely 2-categories and double categories, together with their weak versions, namely bicategories and weak double categories…

The following sections are included:

• References
• Index