An example of universal mapping property appearing
in the definition of terminal object: 1 is terminal
means that for each object X, there is exactly one
map X -> 1. The object 1 is described by its relation
to every object in the ‘universe’, i.e. the category
under consideration.
The idea of figure arises when, in investigating some category C, we find a small class A of objects in C which we use to probe the more complicated objects X by means of maps A ->(x)-> X from objects in A. We call the map x a figure of shape A in X. (or sometimes a singular figure of shape A in X, if we want to emphasize that the map x may collapse A somewhat, so that the picture of A in X may not have all the features of A).
If the categroy C has a terminal object, we can consider it as a basic shape for figures. Indeed, we have already given figures of that shape a special name: a figure of shap 1 in X, 1 -> X, is called a point of X. In sets, the points of X are in a sense all there is to X, so that we often use the words ‘point’ and ‘element’ interchangeably, whereas in dynamical systems points are fixed states, and in graphs they are loops.
The category of sets has a special property, roughly because the objects have no structure: If two points agree on points, they are the same map. That is, suppose X ->(f)-> Y and X ->(g)-> Y, if fx = gx for every point 1 ->(x)-> X, we can conclude that f = g. This special property of the category of sets is not true of S[endomap], nor of S[irreflexive]. In S[endomap] the 2-cycle C2 has no points at all, since ‘points’ are fixed points; any two maps from C2 to any system agree at all points (since they are no points to disagree on) even though they may be different maps.
Given any pair of maps X[endomap:alpha] =>(f,g)=> Y[endomap:beta] in S[endomap], if for all figures N[endomap:delta] ->(x)-> X[endomap:alpha] of shape N[endomap:delta] it is true that fx = gx, then f = g.
Incidence relations
Suppose we have in X a figure x of shape A, and a figure y of shape B. We use a map u: A -> B satisfying yu = x to describe the structure of the overlap.
One way in which x may be incident ot y is if there is map u such that yu = x, another possibility is that we may have maps from an object T to A and to B so that with xu1 = yu2. The second possibility means in effect that there is given a third figure T -> X together with incidences in the first sense to each of x and y.
Basic figure-types, singular figures, and incidence, in the category of graphs
In the category of graphs S[irreflexive] the two objects D = *, and A = * -> * can serve as basic figure-types. A is an arrow of the graph and D is a dot.
Given any pair of maps X ->(f)-> Y, X ->(g)-> Y in S[irreflexive], if fx = gx for all figures D ->(x)-> X of shape D and for all figures A ->(x)-> X of shape A, then f = g.
Another useful figure-type is that of the graph M = * -> * <- *. This graph has two arrows, which means that there are two different maps from A to M, namely the maps.
More on universal mapping properties
Universal mapping properties
| Initial object | Terminal object |
| Sum of two objects | Product of two objects |
| … | Exponential, power, map space (note mentioned yet) |
The definition for a ‘left column property’ is similar to that of the corresponding ‘right column property’ with the only difference that all the maps appearing in the definition are reversed - domain and codomain are interchanged.
Let’s clarify it with a simple example. The idea of initial object is similar to that of terminal object but ‘opposite’. T is a terminal object if for each object X there is exactly one map from X to T, X -> T. Correspondingly, I is an initial object if for each object X there is exactly one map from I to X, I -> X.
In the category of abstract sets an initial object is an empty set.
And, the dual of product of two objects is the sum of two objects.
Round P269 there is an example which gives a construction reduces products in one category to terminal objects in another. (in particular, makes the uniqueness theorem for products a consequence of the corresponding theorem for terminal objects).
Calculate products
We refer to any product B1 <-(p1)<- P ->(p2)->B2 as the product of B1 and B2 and denote the object P by B1 x B2. We call the two maps p1, p2 ‘the projections of the product to its factors’. For any two maps from an object A to B1 and B2, i.e. any B1 <-(f1)<- A ->(f2)-> B2, there is exactly one map A ->(f)-> B1 x B2 satisfying p1f = f1 and p2f = f2. This map is also denoted by a special symbol <f1, f2> which indicates the list of the maps that give rise to f.
Definition: For any pair of maps B1 <-(f1)<- A ->(f2)-> B2, <f1, f2> is
the unique map A -> B1 x B2 that satisfies the equations
p1<f1, f2> = f1 and p2<f1, f2> = f2.
These equations can be read: ‘the first component of the map <f1, f2> is f1’ and ‘the second component of the map <f1, f2> is f2’.
This means, in terms of figures, that the figures of shape A in the product B1 x B2 are precisely the ordered pairs consisting of a figure of shape A in B1 and a figure of shape A in B2. On the one hand, given a figure of shape A in the product B1 x B2, A -> B1 x B2, we obtain figures in B1 and B2 by composing it with the projections; on the other hand, the definition of product syas that any two figures f1 of shape A in B1 and f2 of shape A in B2 arise this way from exactly one figure of shape A in B1 x B2, which we called <f1, f2>.
For any object X in any category having products there is a standard map X -> X x X, namely the one whose components are both the identity map 1X. This standard map is often called the diagonal map.
