ONT Re: Category Theory
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CAT. Note 13
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| 1.2. Categories (cont.)
|
| Discrete Categories. A category is 'discrete' when every arrow
| is an identity. Every set X is the set of objects of a discrete
| category (just add one identity arrow x -> x for each x in X),
| and every discrete category is so determined by its set of
| objects. Thus, discrete categories are sets.
|
| Monoids. A monoid is a category with one object. Each monoid is thus
| determined by the set of all its arrows, by the identity arrow, and
| by the rule for the composition of arrows. Since any two arrows
| have a composite, a monoid may then be described as a set M with
| a binary operation M x M -> M which is associative and has an
| identity (= unit). Thus a monoid is exactly a semigroup with
| identity element. For any category C and any object a in C,
| the set hom(a, a) of all arrows a -> a is a monoid.
|
| Groups. A group is a category with one object in which
| every arrow has a (two-sided) inverse under composition.
|
| Matrices. For each commutative ring K, the set Matr_K of
| all rectangular matrices with entries in K is a category;
| the objects are all positive integers m, n, ..., and each
| m x n matrix A is regarded as an arrow A : n -> m, with
| composition the usual matrix product.
|
| Sets. If V is any set of sets, we take Ens_V to be the category
| with objects all sets X in V, arrows 'all' functions f : X -> Y,
| with the usual composition of functions. By Ens we mean any one
| of these categories.
|
| Mac Lane, 'Cat Work Math', p. 11.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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