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Julius (Jul 16 2023 at 11:16):“a natural transformation is a 2-morphism between two functors.”.
I currently understand a category to be “a collection of objects and a collection of arrows between pairs of objects”. Thus, a category is a directed graph, EXCEPT for one condition, I believe, that of composition of arrows: if a • b and b • c exist, then there necessarily exists a • c too. (cont. below)
Julius (Jul 16 2023 at 11:18):Intuitively, I have been considering something like cooking, colors, or musical tones. An arrow from A to B perhaps is a transformation on an object in some way? For example, let’s say you begin with butter, then add sugar, then add eggs. Those could be arrows, could they not? And the “closure” of the arrows means simply that there is just also one arrow that signifies “add eggs and sugar at the same time”. Is there a different intuition that might be better - like, arrows not as the addition of an element, but a change, like rotations of a geometric object? (cont. below)
Julius (Jul 16 2023 at 11:20):So, let us say, Question 1: what is the true significance of this condition? What important properties does it bring? Without it, I believe we have quasi categories, which still have their own developed theory, right?
Julius (Jul 16 2023 at 11:22):Thus, a natural transformation is perhaps trivially, a functor between functors - does that imply we need to ensure that the functors act like a category? They also need the “every diagram commutes” property, right?
Julius (Jul 16 2023 at 11:25):“A functor is a homomorphism between categories” - a homomorphism is a structure-preserving map as in abstract algebra. It means it respects the relationship the arrows have to each other. I think the arrows of a category form either a monoid or a semigroup.
Julius (Jul 16 2023 at 11:26):I already must clean up my thoughts on all this but many questions have been opened in my mind so I will welcome any conversation partners in an aside-thread where I would not be spamming. Sorry for my beginners-level questions. Thanks for understanding.
John Baez (Jul 16 2023 at 15:13):Welcome, @Julius
“a natural transformation is a 2-morphism between two functors.”
By the way, that's not a definition of natural transformation. It just a fact about natural transformations... and it's not a very helpful fact unless you know what 2-morphisms are.
In math it's often good to start with definitions, and then lots of examples illustrating the definitions. Without knowing definitions and examples, your understanding will be vague. To see the definition of natural transformation, you can do something like go to Wikipedia:
John Baez (Jul 16 2023 at 15:15):But before you understand natural transformations you need to understand functors. And before you understand functors you need to understand categories.
I currently understand a category to be “a collection of objects and a collection of arrows between pairs of objects”. Thus, a category is a directed graph, EXCEPT for one condition, I believe, that of composition of arrows: if a • b and b • c exist, then there necessarily exists a • c too.
This is part of the definition of a category, but it's not the whole definition! There's more, and what you're leaving out is very important. So again: go to some trustworthy source like Wikipedia and read the definition:
Thus, a natural transformation is perhaps trivially, a functor between functors.
No, not true.
“A functor is a homomorphism between categories”
This is a kind of vague fact about categories, not the definition. So again, I urge you to read the definition:
But again, the right sequence is
since each one relies on the previous ones.
Ralph Sarkis (Jul 16 2023 at 15:16):Your intuition about morphisms (seeing them as steps in a recipe) is good. A lot of categories of interest are so-called categories of processes, where objects are states and morphisms are processes that change the state. Your intuition about composition is not totally accurate. Composing "add eggs" with "add sugar" gives "add eggs then add sugar" which is not the same thing as "add eggs and sugar at the same time". Both take you from an empty bowl to a bowl with eggs and sugar, but the processes/morphisms are different.
Ralph Sarkis (Jul 16 2023 at 15:21):Ralph Sarkis said:
Your intuition about morphisms (seeing them as steps in a recipe) is good.
But morphisms can be many many different things, so this intuition can sometimes break. Seeing lots of examples of categories will help you polish your intuition.
Jencel Panic (Jul 16 2023 at 23:15):Welcome @Julius
This diagram might be helpful for you, if you are into cooking:
Note that this is a string diagram, one in which objects are represented by arrows and morphisms are points.
The diagram is from this book: https://arxiv.org/abs/1803.05316, second chapter, I think.
Notification Bot (Jul 17 2023 at 10:11):12 messages were moved here from #general > Introduce yourself! by Morgan Rogers (he/him).
Julius (Jul 22 2023 at 04:41):Thank you. I am actually now reading that book and loving it. It is perfectly at my level. I’d like to open a thread where I discuss and review what I’ve learned from it so far. It is such a masterpiece. It allows you to enter category theory without the advanced background in abstract algebra other texts often seem to require. I’m very excited to read more.
Jencel Panic (Jul 23 2023 at 06:13):Do it, it's a wonderful (the only?) way to learn.
Jencel Panic (Jul 23 2023 at 06:14):There is also the book Category Theory for Scientists by the same author, which is an even better first book https://arxiv.org/pdf/1302.6946.pdf
Julius (Jul 23 2023 at 13:07):Oh shoot, you’re the person who invited me to this Zulip from Twitter! Cool. Thx
Julius (Jul 23 2023 at 13:09):B75673CB-2C1C-4BB3-9456-C13283E77353.png I’d be happy to read your book and give feedback on what parts I find easy vs hard to understand
Jencel Panic (Jul 23 2023 at 19:25):Cool, my book is based on Spivak's work, mostly, btw.
Xuanrui Qi (Jul 31 2023 at 21:07):Or, rather, think of it as a way how A relates to B! There is the standard category of sets and binary relations. Anotjer important example is from symplectic geometry: there is a notion called a Fukaya category, where (roughly) the objects are certain ("Lagrangian") submanifolds of a manifold and the morphisms are (very vaguely) the ways how those submanifolds intersect.