Category Theory - 2017-11-09

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user131753

06:36

Is it possible to do category theory without any prior knowledge of set theory? I am new to this subject and I was wondering the use of set (or class) theory even at the fundamental level to the subject which I thought is expected to serve as an alternative foundation of mathematics.

Martin Sleziak

11:40

I hope that somebody who know more about this topic notices the above question

My personal impression is that if you are studying category theory for the first time, it's better not to worry that much on the issues of size. It is certainly helpful to know that there is a difference between sets and classes, but the most important stuff you're learning is not about such distinctions.

If you wish, maybe you can have a look what Borceux says about this in his handbook, he devotes the first few pages to this issue. And probably any decent category theory textbook will mention something about such issues at the beggining.

I do not know much about foundations based on category theory, so I leave this to others. Certainly, on this site you have some people who knows about this stuf. A quick search returns a few questions which seem to be related to your inquiry.

Is it possible to formulate category theory without set theory?

Category theory without sets

Categorical foundations without set theory

Martin Sleziak

12:11

Sorry for answering this only by links and references - but as I said, I do not know much about foundational issues in category theory, so the best I was able to do was at least trying to direct you to some places where more knowledgeable people said something about this.

Long

12:49

@user170039 In addition to Martin's links: maybe look at Lawvere and Rosebrugh's "Sets for Mathematics", which builds up set theory in category language -- or so I think, because I haven't read it yet (^_^)'...

user131753

13:22

@MartinSleziak: I have found your links to be extremely valuable (especially YonedaLemma's answer). Thank you very much for sharing that.

user131753

How can I draw diagrams in the presheaf.com and share it here?

Martin Sleziak

@user170039 Look at examples there. If you know xypic syntax, it's relatively easy. But we should probably move discussion about this into MathJax chatroom. (So that we do not leave too many off-topic comments here.)

user131753

@MartinSleziak Feel free to do so.

user131753

13:39

Sorry, @Long. I didn't notice that you are the only room owner. In that case my last comment is addressed to you.

Martin Sleziak

I think it's ok if the above messages stays here - what I meant is to continue the MathJax discussion elsewhere and leave this mainly for CT-related discussions. (Which is what we did.) I simply did not want to leave here too many messages which are not directly related to the room topic.

9 hours later…