Subobjects
Definition: In any category, a map S ->(i)-> X is an inclusion, or monomorphism, or
monic map, if it satisfies: For each object T and each pair of maps s1, s2 from T to S
i◦s1 = i◦s2 implies s1 = s2.
Other names for ‘inclusion’ or ‘inclusion map’ are: ‘monic map’ and ‘non-singular map’, or, especially in sets, ‘injective map’ and ‘one-to-one map’. There is a special notation to indicate that a map S ->(i)-> X is an inclusion; instead of writing a plain arrow like -> one put a little hook on its tail, so that S c->i-> X indicates that i is an inclusion map.
Truth
In the category of sets the two-element set 2 = {true, false} has the following property; If X is a set and S c->i-> X is any subset of X, there is exactly one function phiS: X -> 2 such that for all x, x is included in S, i if and only if phiS(x) = true.
Given a part of a set X, say S c->i-> X, its corresponding map X ->phiS-> 2 is called the characteristic map of the part S, i, since at least phiS characterizes the points of X that are included in the part S, i as those points x such that phiS(x) = true.
Actually, the map phiS does much better than that since this characterization is valid for all kinds of figures T ->(x)-> X and not only for points; the only difference is that when T is not the terminal set, we need a map trueT from T to 2 rather than the map true: 1 -> 2. The map trueT is nothing but the composite of the unique map T -> 1 with true: 1 -> 2.
The fundamental property of the characterisitc map phiS is that for any figure T ->(x)-> X, x is included in the part S, i of X if and only if phiS(x) = trueT.
The true value object
P358 some examples for graphs
