Category theory, generalist preferences, and aesthetic value

[Abstract_Logic]

I've just begun learning about category theory. I would love to share ideas with anyone else interested in the topic. Perhaps anyone can recommend some good learning material for it? I've started an introductory book on it, a rather small book of about 200-250 pages. I've finished the first section on general categories, subcategories, pre-categories, morphisms, and other things.

 

I enjoy studying things at the most general level, like category theory, model theory, universal algebra, metamathematics; it intrigues me. I'm more of a generalist than a practitioner, and personally, being a man of pure ideas, I don't believe in anything merely because of its practical value, but rather because of its aesthetic value.

 

One of my favorite quotes: And it is a mistake to think that a mathematical idea can survive merely because it is useful, because it has practical applications. On the contrary, what is useful varies as a function of time, while “a thing of beauty is a joy forever” (Keats). Deep theory is what is really useful, not the ephemeral usefulness of practical applications! - Gregory Chaitin, mathematician and computer scientist

[ajb]

I like category theory, not that I am well versed in it. I am principally a geometer and mathematical physicist, I use basic ideas from category theory all the time. I find it a very economical framework to make clear definitions and statements. I have not really needed to look deeply into category theory like limits etc.

 

Calling category theory "general abstract nonsense", is unfair but common. (Steenrod the first to say this?) However, at first glance it does seem too abstract to be of any real use. It almost feels empty. I do not really believe this is the case, but I am not informed enough to counter this point of view.

 

The most useful thing in category theory is the Yoneda lemma and the notion of a natural transformation.

 

Something I am trying to get a basic acquaintance with now are multicategories and operads. Just some background information really.

 

As for books. The only one I know personally is Geroch [1]. The books is an introduction to many basic structures found in mathematical physics. The interesting thing with this book is that it approaches it using category theory. Something not found in standard text books.

 

Another book I have not really used, but get recommended all the time is MacLane[2].

 

Harold Simms from Manchetser has some notes based on a course he gives. (I attended this course a couple of years ago) You can find it here.

 

 

As for model theory and similar you are now moving outside of what I know.

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[1]R. Geroch, (1985). Mathematical Physics (Lectures in Physics). University of Chicago Press. ISBN 0226288625

 

[2]Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer-Verlag.