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.
Bernd Losert (Dec 05 2023 at 21:39):Is there a reference with a list of categories and all the known (E,M)-factorizations that they have?
Evan Patterson (Dec 05 2023 at 21:43):I have never seen anything that claims to be comprehensive, but Joyal's page on factorization systems on the nLab includes lots of lore, as well as a big list of examples at the end: https://ncatlab.org/joyalscatlab/published/Weak+factorisation+systems
Bernd Losert (Dec 05 2023 at 21:46):I should qualify that I am interested in what some people call "orthogonal" factorizations.
Evan Patterson (Dec 05 2023 at 21:48):Ah, then the better link is this one, which also has lots of examples: https://ncatlab.org/joyalscatlab/published/Factorisation+systems
Bernd Losert (Dec 05 2023 at 21:50):Yes, I was looking at that one. Thanks. The "Joy of Cats" book has a decent amount of examples. I was just wondering if there is something more comprehensive.
fosco (Dec 06 2023 at 11:02):Listing "all" factorization systems on a category is hardly something feasible, in a sense similar to the fact that listing "all" subsets of the real line is unfeasible...
You can study classes of factorization systems, or assumptions under which well-behaved (for example, generated by a set of bounded cardinality) FS on a category arise.
Bernd Losert (Dec 06 2023 at 21:39):Agreed. I did not literally mean "all" factorization systems, just the ones that are commonly known.
fosco (Dec 06 2023 at 22:14):Your question is tightly related to this one (it's not the same thing, but every prefactorization system of a certain shape, where the prefix "pre-" means just a pair of mutually orthogonal subcategories, without the property that each arrow factors as , arises in this way).
Take a category and try to classify all reflective subcategories of . Clearly, without a concrete definition of , allowing ad-hoc arguments, and without some constraint on which reflections you want to study, there is little you can say.
Mike Shulman (Dec 06 2023 at 22:18):However, you can list literally all the factorization systems on the category of sets.