ONT Re: Category Theory
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CAT. Note 8
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| Excerpts from 'Categories for the Working Mathematician' by Saunders Mac Lane
|
| 1. Categories, Functors, and Natural Transformations
|
| 1.1. Axioms for Categories
|
| First we describe categories directly by means of axioms,
| without using any set theory, and call them "metacategories".
| Actually, we begin with a simpler notion, a (meta)graph.
|
| A 'metagraph' consists of:
|
| 'objects' a, b, c, ...,
|
| 'arrows' f, g, h, ...,
|
| and two operations, as follows:
|
| 'Domain', which assigns to each arrow f an object a = dom f.
|
| 'Codomain', which assigns to each arrow f an object b = cod f.
|
| These operations on f are best indicated by displaying f
| as an actual arrow starting at its domain (or "source")
| and ending at its codomain (or "target").
|
| f
| f : a -> b or a ---> b.
|
| A finite graph may be readily exhibited:
|
| --->
| Thus o--->o--->o or o o
| --->
|
| A 'metacategory' is a metagraph with two additional operations:
|
| 'Identity',
|
| which assigns to each object 'a'
| an arrow id_a = 1_a : a -> a.
|
| 'Composition',
|
| which assigns to each pair <g, f> of arrows with
| dom g = cod f an arrow g o f, called their 'composite',
| with g o f : dom f -> cod g. This operation may be
| pictured by the diagram:
|
| b
| o
| ^ \
| / \
| f / \ g
| / \
| / v
| a o---------->o c
| g o f
|
| which exhibits all domains and codomains involved.
|
| These operations in a metacategory are subject to the two following axioms:
|
| 'Associativity'.
|
| For given objects and arrows in the configuration:
|
| f g k
| a ---> b ---> c ---> d
|
| one always has the equality:
|
| k o (g o f) = (k o g) o f. (1)
|
| This axiom asserts that the associative law holds for
| the operation of composition whenever it makes sense (i.e.,
| whenever the composites on either side of (1) are defined).
| This equation is represented pictorially by the statement
| that the following diagram is commutative:
|
| k o (g o f) = (k o g) o f
| a o-------------------------->o d
| | . ^ |
| | . g o f k o g . |
| | . . |
| | . . |
| | . . |
| | . . |
| f | . | k
| | . . |
| | . . |
| | . . |
| | . . |
| | . . |
| v . v |
| b o-------------------------->o c
| g
|
| 'Unit law'.
|
| For all arrows f : a -> b and g : b -> c
| composition with the identity arrow 1_b gives:
|
| 1_b o f = f and g o 1_b = g. (2)
|
| This axiom asserts that the identity arrow 1_b of each object b
| acts as an identity for the operation of composition, whenever
| this makes sense. The Eqs. (2) may be represented pictorially
| by the statement that the following diagram is commutative:
|
| f
| a o-------->o b
| \ |\
| \ | \
| \ | \
| \ | \
| f \ 1_b \ g
| \ | \
| \ | \
| \ | \
| vv v
| b o-------->o c
| g
|
| We use many such diagrams consisting of vertices (labelled by objects
| of a category) and edges (labelled by arrows of the same category).
| Such a diagram is 'commutative' when, for each pair of vertices
| c and c', any two paths formed from directed edges leading from
| c to c' yield, by composition of labels, equal arrows from
| c to c'. A considerable part of the effectiveness of
| categorical methods rests on the fact that such
| diagrams in each situation vividly represent
| the actions of the arrows at hand.
|
| If b is any object of a metacategory C, the corresponding identity arrow
| 1_b is uniquely determined by the properties (2). For this reason, it is
| sometimes convenient to identify the identity arrow 1_b with the object b
| itself, writing b : b -> b. Thus 1_b = b = id_b, as may be convenient.
|
| Mac Lane, 'Cat Work Math', pp. 7-8.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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