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By Shankar Venkataraman

If marriage is the legally binding union of family, is it reasonable to claim marriage in accordance with prearranged phrases is a tainting of affection in itself? As resident of a society during which the proposal of a love established marriage is idealized, we need to commence wondering if prepared marriages faded compared to love marriages or even if prearranged unions are even more appropriate to fashionable day international electorate than age outdated romanticism.

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This ambiguity will only be fully elucidated once we establish that, ultimately, category theory is essentially an intuitionistic logic (as indeed is Cartesian logic). Which is to say that it organizes thought around actual constructions, which can in effect either be explicit equations or ‘spatial’ configurations. We will say that a diagram is commutative if all its triangles are commutative. Since the problem of the size of a category is that of the ‘perspective’ that we can have with regard to a diagram, it is simply a question of defining the metaphor of ‘seeing’, or of perspective, consistently, thus with consistent diagrams.

Now, let’s suppose that this diagram is a pullback. Or to put it another way, the object a and the arrow Id(a) are limited by the diagram: This means that for every object c which admits arrows g and h toward a and the appropriate triangular commutations, we get the following diagrammatic situation: Given that the diagram commutes, we have Id(a) ◦ k = h. But by the definition of Id(a) (null action), this means that k = h. In the same way (the other triangle of the commutative triangle) we have Id(a) ◦ k = g, therefore k = g.

Such an object, and the system of actions (or arrows) which identify it, will receive a conceptual name. A category will be gradually described by the existence (or not) of objects that fall under such concepts. In particular, the greater the number of limits for every kind of diagram, the ‘larger’ it will be. The extension acquires a descriptive or intentional sense (to support a wide range of conceptual names, or to provide, from within the category, the cases for these names), and not, as in set theory, a directly extensional sense (the cardinality of the universe).

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