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November 15, 2012

The trouble with singletons is that there is only one function to 1 = {.}.  If we have a domain set D = {d1, d2}, then both elements will be assigned one and the same value ‘.’ by the only function

f: D -> 1

We see this troubling behavior even in those whose last name is not 1

f: D -> C

assigning both elements d1, d2 of D to one element, say, c1 of the codomain C = {c1, c2}.  If we can’t use their codomains to weed-out these guys, then find some way to profile them!  (Oops! sorry, I didn’t mean to sound so grating.)

This troublesomeness has been recognized as a constant map–a map

f: D -> C

that can be factored through a terminal object T


g: D -> T, h: T -> C

And these terminal objects are all one of those kinds: objects satisfying universal mapping properties.  So we might as well round up all those guys: for every map

f: D -> C

get me all those maps from domain object D to objects satisfying universal mapping properties along with those to codomain object C from objects satisfying universal mapping properties.

You crazy-man?  That’s like half-the-continent!  Take a deep breath, exhale slowly … how about I give you a function

f: D -> C

D = {d1, d2}, C = {c}

and you investigate to your heart’s-content with

g: D -> DCD, h: DCD -> C

Go easy i.e. start with

f2 -> 1

g2 -> 2h2 -> 1

This is red-tape!  Bureaucracy-in-action :-(

Dealing with the cards dealt: we might find something interesting (or so it seems listening to comforting thoughts); after all we are talking, given a function

f: D -> C,

about functions with the domain set of f as domain

g: D -> OUMP

and about functions with the codomain set of f as codomain

h: OUMP -> C

and, given the common (to functions g and h) set OUMP, about triangles (and their commutativity) with fg, and h as sides, and in one last one and all the attedant allusions to being and existence [in alphabetical order] are making Zzz

Exercise 1, Conceptual Mathematics, page 306

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