ONT Re: Inquiry Into Irreducibility
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I omitted a figure, which left its lead-in
with nothing to lead up to. Here is a fix:
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We have just illustrated the circumstance -- coincidence? I think not! --
that a deux-tuple is determined by the details of its projective pieces --
still, according to a sage bit of wisdom, if a deux-tuple did not exist
that was determined by the data of its projective parts, then it should
be necessary to invent one that was -- which is parably what this bible
of pan-categorics tells us about the genesis of every cartesian product:
| Many properties of mathematical constructions may
| be represented by universal properties of diagrams.
| Consider the cartesian product X x Y of two sets,
| consisting as usual of all ordered pairs <x, y>
| of elements x in X and y in Y. The projections
| <x, y> ~> x, <x, y> ~> y of the product on its "axes"
| X and Y are functions p : X x Y -> X, q : X x Y -> Y.
| Any function h : W -> X x Y from a third set W is uniquely
| determined by its composites p o h and q o h. Conversely,
| given W and two functions f and g as in the diagram below,
| there is a unique function h which makes the diagram commute;
| namely, hw = <fw, gw>:
|
|
| W
| o
| /|\
| / | \
| / | \
| f / h \ g
| / | \
| / | \
| / | \
| v v v
| o<---p---o---q--->o
| X X x Y Y
|
|
| Thus, given X and Y, <p, q> is "universal" among pairs of
| functions from some set to X and Y, because any other such
| pair <f, g> factors uniquely (via h) through the pair <p, q>.
| This property describes the cartesian product X x Y uniquely
| (up to a bijection); the same diagram, read in the category
| of topological spaces or of groups, describes uniquely the
| cartesian product of spaces or the direct product of groups.
|
| Mac Lane, 'Cat Work Math', pages 1-2.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| Springer-Verlag, New York, NY, 1971.
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