ONT Higher Order Categorical Logic
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Higher Order Categorical Logic
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Note 1
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| 1. Categories and Functors
|
| In this section we present what our reader is expected
| to know about category theory. We begin with a rather
| informal definition.
|
| Definition 1.1. A 'concrete category' is a collection of two kinds
| of entities, called 'objects' and 'morphisms'. The former are sets
| which are endowed with some kind of structure, and the latter are
| mappings, that is, functions from one object to another, in some
| sense preserving that structure. Among the morphisms, there is
| attached to each object A the 'identity mapping' 1_A : A -> A
| such that 1_A(a) = a for all a in A. Moreover, morphisms
| f : A -> B and g : B -> C may be 'composed' to produce
| a morphism gf : A -> C such that (gf)(a) = g(f(a))
| for all a in A.
|
| Examples of concrete categories abound in mathematics;
| here are just three:
|
| Example C1. The category of 'sets'. Its objects are
| arbitrary sets and its morphisms are arbitrary mappings.
| We call this category "Sets".
|
| Example C2. The category of 'monoids'. Its objects are
| monoids, that is, semigroups with unity element, and its
| morphisms are homomorphisms, that is, mappings which
| preserve multiplication (the semigroup operation)
| and the unity element.
|
| Example C3. The category of 'preordered sets'.
| Its objects are preordered sets, that is, sets
| with a transitive and reflexive relation on them,
| and its morphisms are monotone mappings, that is,
| mappings which preserve this relation.
|
| The reader will be able to think of many other examples:
| the categories of rings, topological spaces, and Banach
| algebras, to name just a few. In fact, one is tempted
| to make a generalization, which may be summed up as
| follows, provided we understand "object" to mean
| "structured set".
|
| Slogan 1. Many objects of interest in mathematics
| congregate in concrete categories.
|
| L&S, page 4.
|
| Lambek, J. & Scott, P.J.,
|'Introduction To Higher Order Categorical Logic',
| Cambridge University Press, Cambridge, UK, 1986.
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