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- Morphism that is not a mapping 3 answers
In my loose reading about category theory, categories are usually introduced in terms of sets, often with some structure (such as algebraic or topological), and functions between them that preserve that structure. I find this easy to intuit, and makes ideas like commutative diagrams easy to understand.
However, I know that categories can be much broader than this, even before we get into higher notions like functors and such, and that categories describing structured sets and the functions between them are only an important but small portion of them. Despite this, I find it hard not to unconsciously interpret diagrams and category theoretic statements in these terms unless I completely strip all meaning from the symbols and treat the whole thing as a purely syntactic game, and the times where I've seen people discussing categories where the morphisms aren't structure preserving functions have been very abstract and gone right over my head.
So what I would like are some simple, intuitive examples of things which can be described by category theory, but where -
A) The objects are not sets,
B) the morphisms aren't (possibly structure preserving) functions, or
I've wanted to learn more about category theory for a while, but I've found I kept being held back by my lack of a strong, accurate mental model for the sorts of things categories can represent, in the way that I have a strong model for the ways sets work and the things they can represent.