Changes

Last time we looked at an ordinary function f : X -> Y, and we glommed onto a single fiber of f, considered in one of two ways:  (1) a set of ordered pairs F c X x Y such that <x, y> in F if and only f(x) = y, or else (2) a subset of X, horrifically asciified as f^(-1)(y) c X.

Last time we looked at an ordinary function f : X -> Y, and we glommed onto a single fiber of f, considered in one of two ways:  (1) a set of ordered pairs F c X x Y such that <x, y> in F if and only f(x) = y, or else (2) a subset of X, horrifically asciified as f^(-1)(y) c X.

<font face="courier new"><pre>

<font face="courier new"><pre>

o-----------------------------o-----------------------------o

o-----------------------------o-----------------------------o

o-----------------------------------------------------------o

o-----------------------------------------------------------o

</pre></font>

</pre></font>

Any function f : X -> Y, which is, after all, exactly the same as the relation f c X x Y, can be treated as an assortment of fibers of the first sort.  So it is easy to grasp the elementary fact of category theory that any function whatsoever can be factored into an epic (surjective, "onto") and a monic (injective, "one to one") sequence of composed functions, as illustrated here for one fiber:

Any function f : X -> Y, which is, after all, exactly the same as the relation f c X x Y, can be treated as an assortment of fibers of the first sort.  So it is easy to grasp the elementary fact of category theory that any function whatsoever can be factored into an epic (surjective, "onto") and a monic (injective, "one to one") sequence of composed functions, as illustrated here for one fiber:

<font face="courier new"><pre>

<font face="courier new"><pre>

o-------------------o-------------------o-------------------o

o-------------------o-------------------o-------------------o

o-----------------------------------------------------------o

o-----------------------------------------------------------o

</pre></font>

</pre></font>

In sum, an arbitrary f : X -> Y can always be factored into a pair of functions of the types g : X -> M and h : M -> Y, where g is surjective, h is injective, and f = h o g, here using the "left-composition" convention according to which the composition h o g is defined by (h o g)(x) = h(g(x)).

In sum, an arbitrary f : X -> Y can always be factored into a pair of functions of the types g : X -> M and h : M -> Y, where g is surjective, h is injective, and f = h o g, here using the "left-composition" convention according to which the composition h o g is defined by (h o g)(x) = h(g(x)).

We began with the trusim from category theory, at least, the sorts of "concrete categories" of sets and functions that will be most salient in the minds of most everybody:  That an arbitrary arrow factors into a couple of pieces, an epic on which a monic ensues, 'Iliad' and 'Odyssey', if you will, and if you catch my drift, and whether you will or not, 'tis true.

We began with the trusim from category theory, at least, the sorts of "concrete categories" of sets and functions that will be most salient in the minds of most everybody:  That an arbitrary arrow factors into a couple of pieces, an epic on which a monic ensues, 'Iliad' and 'Odyssey', if you will, and if you catch my drift, and whether you will or not, 'tis true.

<font face="courier new"><pre>

<font face="courier new"><pre>

|                  f

|              arbitrary

|        X o-------------->o Y

|                  f                   |

|            \            ^

|              arbitrary               |

|            \          /

|        X o-------------->o Y         |

|      g      \        /    h

|            \            ^           |

|  surjective  \      /  injective

|            \          /             |

|    "epic"    \    /    "monic"

|      g      \        /    h         |

|                \  /

|  surjective  \      /  injective   |

|                  v /

|    "epic"    \    /    "monic"     |

|                  o

|                \  /                 |

|                  M

|                  v /                 |

|                  o                   |

|                  M                   |

</pre></font>

</pre></font>

Now, there's a catch here -- there's always a catch, the way I see it -- leastwise, once we begin to think so systematically as to be working inside any sort of category at all, instead of merely picking up on this or that isolated instance of an arbalistrary functional arrow, then this ostensibly trivial truism becomes contingent on the list of a "suitable transitional object" (STO), like M in our example, and of the "requisite intermedi-arrows" (RIA's), like g and h, explicitly listed within the formal category in question.  Otherwise, "you just cannot get there from here" is the only thing that answers to your desire for mediation.

Now, there's a catch here &mdash; there's always a catch, the way I see it &mdash; leastwise, once we begin to think so systematically as to be working inside any sort of category at all, instead of merely picking up on this or that isolated instance of an arbalistrary functional arrow, then this ostensibly trivial truism becomes contingent on the list of a "suitable transitional object" (STO), like M in our example, and of the "requisite intermedi-arrows" (RIA's), like g and h, explicitly listed within the formal category in question.  Otherwise, "you just cannot get there from here" is the only thing that answers to your desire for mediation.

===Incidental Note 10===

===Incidental Note 10===