Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Study order for concepts in category theory

David Tanzer (Sep 18 2026 at 17:28):

How would one recommend an order for learning concepts in category theory, based on their fundamental significance, and their progression as conceptual building blocks.

Something like: category, functor, natural transformation, colimit / limit, adjunction, ... ?

Any suggestions could be helpful. Either partial orders, or lists of concepts. Thanks.

David Tanzer (Sep 18 2026 at 17:33):

with a side-chain for monoidal category, SMC, ...

David Tanzer (Sep 18 2026 at 17:51):

special functors: forgetful, free-generating, ...

David Tanzer (Sep 18 2026 at 17:56):

chain: Hom-functor, Yoneda embedding, ...

early concept: quiver

David Tanzer (Sep 18 2026 at 18:02):

complementary perspective: categories as partial algebras of arrows

Ben Kaminsky (Sep 18 2026 at 18:37):

I would probably just suggest reading Leinster --- Basic Cat. Theory. The slope/ordering is more or less what I think you are looking for.

David Tanzer (Sep 18 2026 at 22:47):

Thanks @Ben Kaminsky ! I ordered it, and look forward to going through it.

John Baez (Sep 19 2026 at 06:49):

I seem to recall that book says nothing about monoidal categories; that's a distinct and more modern strand of category theory (starting in 1963 rather than 1945-ca 1955). For that you might try Noson Yanofsky's book on monoidal categories, very expository and packed with examples. But that can come later, if you want. Leinster's book is good for the basic stuff. If Leinster is too hard, you can try Cheng's The Joy of Abstraction, which does categories, functors and natural transformations, up to the Yoneda lemma, with lots of important examples, and lots of the associated ways of thinking spelled out.

David Tanzer (Sep 19 2026 at 18:11):

Thanks for the recommendations John! The Yanofsky book sounds like a great complement to Leinster - I just ordered a copy.

Paolo Perrone (Sep 21 2026 at 10:21):

If you come from math/theoretical physics I'd do:
1) categories, functors, natural transformations
2) limits and colimits
3) monoidal categories
4) monads
5) adjunctions
6) Yoneda/universal properties in general
7) some basic enrichment (e.g. abelian categories if you're interested in cohomology)

and then go towards any advanced or special topic you need

David Tanzer (Sep 21 2026 at 21:19):

Thanks Paolo! That's quite helpful. Good to see monads on the schedule. I'm orienting towards applications of monoidal categories in computer science. One brick at a time, though.

John Baez (Sep 22 2026 at 11:39):

I learned monads after monoids in monoidal categories, so "a monad is just a monoid in a monoidal category of endofunctors" is not a joke to me. But I'm not sure I know a textbook that teaches things in @Paolo Perrone's desired order. Maybe Fong--Spivak???

Paolo Perrone (Sep 22 2026 at 12:03):

My book is not written in that order, but it can be more or less studied in that order.
Seven Sketches is a little more like that, yes.
I also second Leinster's book very much!