Binary operations and actions
We will study two cases of mapping a product to an object. The first case is that in which the three objects are the same, i.e. maps B x B -> B. Such a map is called a binary operation on the object B. The word ‘binary’ refers to the fact that an input of the map consists of two elements of B. (A map B x B x B -> B is a ternary operation on B, and unary operations are the same as endomaps).
Another case of a mapping with domain a product is a map X x B -> X. Such a map is called an action of B on X.
Cantor’s diagonal argument
The most general case of a map whose domain is a product has all three objects different: T x X ->(f)-> Y. Again each point 1 ->(x)-> X yields a map T ->(f(-,x))-> Y, so that f gives rise to a family of maps T -> Y, one for each point of X, or as we often say, a family parameterized by (the points of) X, in this case a family of maps T -> Y. In the category of sets for each given pair T, Y of sets, there is a set X big enough so that for an appropriate single map f, the maps f(-, x) give all maps T -> Y, as x runs through the points of X.
One might think that if T were infinite, we would not need to take X bigger; however it is wrong, the famous theorem proved over one hundred years ago by Georg Cantor shows: T itself (infinite or not) is essentially never big enough to serve as the domain of a parameterization of all maps T -> Y.
Diagonal Theorem: (In any category with products) If Y is an object such that
there exists an object T with enough points to parameterize all the maps T -> Y by
means of some single map T x T ->(f)-> Y, then Y has the ‘fixed point property’: every
endomap Y ->(alpha)-> Y of Y has at least one point 1 ->(y)-> Y for which alpha◦y = y.
Cantor's Contrapositive Corollary: If Y is an object known to have at least one endomap
alpha which has no fixed points, then for every object T and for every attempt f: T x T -> Y
to parameterize maps T -> Y by points of T, there must be at least one map T -> Y which
is left out of the family, i.e. does not occur as f(-, x) for any point x in T.
