ONT Re: Category Theory
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CAT. Note 12
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NB. Mac Lane uses a symbol for the one object and one (identity) arrow
category that looks like a dot with a sling out of it and an arrow
back into it. I will use a "Greek amphora" or "emphattic at" sign
for this, like so "!@!".
| 1.2. Categories (cont.)
|
| For the moment, we consider examples.
|
| $0$ is the empty category (no objects, no arrows).
|
| $1$ is the category !@! with one object and one (identity) arrow.
|
| $2$ is the category !@! -> !@! with two objects a, b,
| and just one arrow a -> b not the identity.
|
| $3$ is the category with three objects whose non-identity arrows
| are arranged as in the triangle [in the "transitive" manner]:
|
| o
| ^ \
| / v
| o---->o
|
| $||$ is the category with two objects a, b and just two
| arrows a -> b not the identity arrows. We call two
| such arrows 'parallel arrows'.
|
| In each of the cases above there is only
| one possible definition of composition.
|
| Mac Lane, 'Cat Work Math', pp. 10-11.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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