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Timothy (Apr 06 2022 at 10:09):Hi All :wave:
I'm new to this group, so a brief introduction is in order. I'm working in bioinformatics, but with a background in maths/stats and some experience with group theory (but only a rough understanding of the basics of category theory).
I'm fed up with compromises with data management, so I want to create a well-designed, robust framework for the language I use for data analysis (Julia). So far, I've written a bit of a manifesto with my thoughts on data management (https://tecosaur.com/public/DataManagement.html). At the end of this, I have a suspicion that my approach might be able to be reformulated as a category theory problem, and in doing so provide a clean abstraction/model for what I'm doing.
Unfortunately, I don't know enough about category theory to take my suspicion any further, which is where I'm hoping you fine folks might be able to help me out :smile:.
Timothy (Apr 06 2022 at 10:10):(I'm guessing this is the right place to post my question/help request, if not please let me know where it should go)
Timothy (Apr 06 2022 at 10:11):In case it's of any help, here's a diagram showing a bit of the nature of my approach:
Jacob Zelko (Apr 06 2022 at 15:46):I cannot comment directly on your approach right now but have you read the excellent paper written by @Evan Patterson @Owen Lynch and @James Fairbanks on Attributed C-Sets? https://arxiv.org/abs/2106.04703
Very practical for applying category theory to code.
Also, be sure to check out AlgebraicJulia here and especially their blog!
Jacob Zelko (Apr 06 2022 at 15:47):P.S. Nice to cross paths again @Timothy :wave:
Timothy (Apr 06 2022 at 15:48):Hello! I recognise that avatar... :thinking:
Thanks for the links, I'll check out that paper.
GhaS Shee (Apr 23 2022 at 03:08):Hello, Timothy!
It might be a bit hardar way to obtain some good intuition around category theory,
but I would like you to enjoy starting them!
In my understanding, Category theory is much depending on Homology theory!
It might help you to try some simple homology theory and then famous books such as
Mac Lane's "category theory for working mathematician" and Emily Riehl's "Category theory in context" !
These books are really general introduction for CT, and really deep such that the generalization would capture any regions of science in a good manner :-)
If you have already been introduced to category theory properly, I hope you will enjoy ∞-category theory by Cisinski's book ! http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf
In applied category theory, there might be lots of works depending on modern cateogry theory and geometric invariances such as homotopy and homology. To see the works of good abstractions, these books would help.
In reality, I have been depending much on Kashiwara Schapira's "categories and sheaves" to understand category theory, but it was a bit elaboration to read this one. However, it gave me great details.
John Baez (Apr 23 2022 at 17:31):One approach to category theory is through algebraic topology and algebraic geometry. This is the historical approach. Eilenberg and Mac Lane invented categories to formalize the notion of "natural isomorphism" between cohomology theories, Grothendieck and others invented abelian categories to help understand sheaf cohomology, later Grothendieck invented topoi to understand sheaves more deeply, Quillen invented model categories to formalize the analogy between chain complexes and topological spaces, and so on.
But by now there are many other routes to category theory! For example, there are a lot of people using category theory in computer science, and such people don't need to understand algebraic topology or algebraic geometry to learn category theory.
There are also a lot of people using category theory to study quantum physics and quantum field theory: that's how I got interested in categories.
And there are people applying category theory to other things. I'll soon be mentoring a group of 14 students on a project about category theory and chemistry, where understanding "Petri nets with rates" is much more important than understanding anything about algebraic topology.
So nobody should feel they need to understand cohomology to learn category theory!
(But of course cohomology theory is fascinating in itself, and I've spent years studying it.)
John Baez (Apr 23 2022 at 17:37):For @Timothy I recommend this free book (if he hasn't already read it):
John Baez (Apr 23 2022 at 17:40):There are also lots of other useful books, including those by Mac Lane, Riehl and Leinster. The latter two are free online - legally, that is.