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Jul. 10th, 2017 12:14 pmAnother interesting quote from McLane&Moerdijk, 1999.
In short, we can create two contrasting categories: a poset P, which has one morphism and many objects, and a group G, which has one object and many morphisms.
Now, by thinking in topos terms, we can see that both categories have a terminal object ( "the arrow", "the object") and can be used in combination with a subobject classifier. The existence of a subobject classifier creates an opportunity for a simple decision making process. Then the question arises, "Decision making about what?"
One way to answer it would be to go back to Adam Smith and say, "Specialization!" That is, posets P enable us to specialize in arrows ("the arrow"), while groups G (including monoids) let us specialize in objects ("the object".) A combination (e.g. a disjoint union) of these two categories helps us build a broad range of large, highly specialized categories with relatively simple decision making rules. I bet most industrial systems can be modeled this way.
In short, we can create two contrasting categories: a poset P, which has one morphism and many objects, and a group G, which has one object and many morphisms.
Now, by thinking in topos terms, we can see that both categories have a terminal object ( "the arrow", "the object") and can be used in combination with a subobject classifier. The existence of a subobject classifier creates an opportunity for a simple decision making process. Then the question arises, "Decision making about what?"
One way to answer it would be to go back to Adam Smith and say, "Specialization!" That is, posets P enable us to specialize in arrows ("the arrow"), while groups G (including monoids) let us specialize in objects ("the object".) A combination (e.g. a disjoint union) of these two categories helps us build a broad range of large, highly specialized categories with relatively simple decision making rules. I bet most industrial systems can be modeled this way.
