Isn't that one of the basis of maths? Namely you choose graph theory or category theory and you can rebuilt the whole maths?
If yes, which one is the "simpler", category or graph?
If yes, which one is the "simpler", category or graph?
Set theory is generally taught before category theory at universities. "Simpler" is very subjective, but set theory is often seen as more intuitive because some of the concepts start getting taught much earlier in school.
I know sets well because well that's what I was taught in school. I'm just trying to understand where set theory fits in category theory or if they are two totally different things.
One key connection is that the category of sets, called "Set," is a fundamental example in category theory. Its objects are sets, and its morphisms are functions between them. This shows how set theory can be seen as a special case of category theory. At the same time, many categories studied in category theory have underlying set structures. For example, groups, rings, or vector spaces are often built on sets with additional structure.
Another connection lies in how category theory generalizes set-theoretic ideas. Concepts like products, coproducts, limits, and colimits in category theory extend the familiar notions of Cartesian products or unions in set theory. The Yoneda Lemma, a cornerstone of category theory, relies heavily on set-theoretic intuition, as it connects abstract categorical concepts to concrete representations in terms of sets.
Topoi, a concept in category theory, also bridge the two fields. A topos generalizes set theory by providing a categorical framework that behaves like the category of sets but with additional logical structure. This allows set-theoretic reasoning to be carried out in a more abstract setting. For instance, the category of sets itself is the prototypical example of a topos.
However, there are also differences. Set theory often grapples with "size" issues, such as distinguishing between sets and proper classes. In category theory, similar concerns arise, especially with large categories, like the category of all sets, which cannot itself be a set. Set-theoretic tools are often used to handle these size issues by defining notions like small and large categories.
And there’s a philosophical aspect. Set theory forms the traditional foundation of mathematics with its ZFC axioms, focusing on elements and collections. Category theory offers an alternative foundation by emphasizing relationships and structures, often referred to as "structural mathematics." These perspectives aren’t in opposition but rather complementary, with each offering tools to understand mathematics from different angles.
In essence, category theory builds on and extends set theory while also relying on it for foundational concepts. The two fields are tightly intertwined, with set theory providing a concrete framework and category theory offering an abstract, flexible way to study and generalize mathematical structures.
[1] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... [2] https://en.wikipedia.org/wiki/Grothendieck_universe
https://www.quora.com/Are-there-any-axioms-for-category-theo...
Category theory is the study of collections of mathematical of a give type. The category of groups, the category of sets, the category of vector spaces. The key facet of category theory is that you can have "functions" (called functors) between categories and the power of category theory is the study of these functors. I put "functions" in parentheses because most categories are not sets in the set-theoretic sense because they do not have a well-defined cardinality. Of course, some categories called "small categories" are sets.
Both category theory and set theory use sets. But set theory tries to make absolutely everything into a set. It takes on a little complexity in this quest, because of Russel's paradox. Category theory can be seen as studying set theory, among other things, so it is "bigger" or more all-encompassing than set theory. Many things studied in category theory aren't possibly sets.
Set theory is more immediately intuitive, but category theory organizes things in a way that are ultimately more insightful, I think. Meaning that once you can get into the category theory headspace and learn to navigate it, it becomes a much better environment for thinking without mistakes. In my personal opinion.