Is Category Theory geometric?

Solution 1:

At the risk of oversimplification: Eddington in "Nature of the Physical World" referring to relativity said "But besides the geometrisation of mechanics there has been a mechanization of geometry."

So in effect he is saying that not only have causal processes formally described through mechanics achieved a more consistent framework by viewing them (geo)metrically but also that (geo)metrical structures can also lead to a consistent framework of causal processes.

If we view this quote in a categorical perspective, then in essence he is saying nothing more than the fact that any process can be reduced to a set of objects and arrows, where those arrows can now represent any mechanical process. On the dual side, we can look to these abstract set of objects and arrows and from them extract universal mechanics, universal mapping properties, such as the principles involved in natural transformations and adjunctions.

It is in this sense, that category theory allows us to identify any causal process (functions, maps, transformations; anything with a "before" and "after") with an equivalent relational structure, a geometry (poset, group, lattice, etc), that I think is at the core of his statement.

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