category theory

As mentioned earlier set theory provides a general conceptual framework for mathematics. Now, since category theory, through the notion of topos, has succeeded in axiomatising set-theory, the outcome is an entirely new categorial foundation of mathematics! The category-theorists attitude that "function" rather than "set membership" can be seen as the fundamental mathematical concept has been entirely vindicated. The pre-eminent role of set theory in contemporary mathematics is suddenly challenged. A revolution has occurred in the history of mathematical ideas (albeit a peaceful one) that will undoubtedly influence the direction of the path to the future.

1984, Robert Goldblatt, Topoi, the categorial analysis of logic, page 3

[Eugenia Cheng] We don't talk about this enough. We make choices about what to count as the same and what to count as different, and we should be aware that we're making them. [Katrina Miller:] What kind of math do you explore in the book? [Cheng 2025: Unequal: The Math of When Things Do and Don't Add Up] [Cheng:] Category theory is a branch of math with this radical idea that when you study things, you don’t have to know what they are. What you have to know is how they interact with other things. It's all about relationships rather than intrinsic characteristics. And one of the most important relationships is sameness and equivalence. When is a relationship good enough to count as things being the same? One of my favorite tongue-in-cheek examples is if you need to temporarily take over someone's role at work because they're on leave. It typically doesn't matter what you look like. Unless the work is being a body double!

2025 August 30, Katrina Miller, “You Don't Need to Be Good at Math to Enjoy It. In her latest book, Eugenia Cheng, a mathematician, explores the choices we make to determine if two things — numbers, shapes, words and even people — are equal. 'A Conversation With' column”, in New York Times