In "From a Geometrical Point of View" (http://www.amazon.com/gp/aw/d/1402093837?pc_redir=1407132421&robot_redir=1) Marquis states that category theory is thoroughly geometric. Could someone explain his use of the term "geometric" here. I would love to read this text, unfortunately the price ($200 plus) is a bit steep for me.

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    $\begingroup$ It was in a geometry course that I first learned about commutative diagrams. (But that comment falls far short of answering this question.) $\endgroup$ – Michael Hardy Aug 14 '14 at 2:35
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    $\begingroup$ If you are a student, you might look to see if this book is available as an ebook through your university library. It was available through mine (alas, it's a temporary file and they'll soon take away my account since I've graduated). You could also try to find a physical copy in a library =) $\endgroup$ – vociferous_rutabaga Aug 14 '14 at 2:50
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    $\begingroup$ This book review claims "in this book his principal objective is to establish the claim that category theory is a generalization of Felix Klein's Erlangen program." So, what the author means by "geometrical" should probably be understood in that light... $\endgroup$ – Alex Nelson Aug 14 '14 at 4:02
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    $\begingroup$ It might also refer to the fact that the theory of categories (as a first-order theory) is geometric. See here. $\endgroup$ – Pece Aug 14 '14 at 7:11
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    $\begingroup$ The book is available at the library of our faculty. If no one else answers, I may skim through it and try to answer the question. $\endgroup$ – Martin Brandenburg Aug 14 '14 at 10:09

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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