Introduction

Discussion 2.1.1 A category consists of a pair of collections, namely a collection of “structures” together with a collection of “relations between the structures.” Let us illustrate this with some informal examples of categories.

• The collection of all sets (thus each set is an example of one of the structures referred to above), together with the collection of all set-theoretic functions (the functions are the relations between the structures).

• The collection of all posets (each poset is a structure), together with all monotone functions (the monotone functions are the relations between the structures).

• The collection of all finite dimensional vector spaces, together with all linear maps.

• The set of real numbers ℝ (in this case each structure is just a real number r ∈ ℝ), together with the relation of order ≤ on the set ℝ. Thus given two structures r, r′ ∈ℝ, there is a relation between them just in case r ≤r′.

All categories have this basic form, that is, consist of structures and relations between the structures: the structures are usually referred to as the objects of the category and the relations between the structures as morphisms. It is important to note that the objects of a category do not have to be sets (in the fourth example they are real numbers) and that the morphisms do not have to be functions (in the fourth example they are instances of the order relation ≤).