https://plato.stanford.edu/entries/category-theory/

Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. Roughly, it is a general mathematical theory of structures and of systems of structures. As category theory is still evolving, its functions are correspondingly developing, expanding and multiplying. At minimum, it is a powerful language, or conceptual framework, allowing us to see the universal components of a family of structures of a given kind, and how structures of different kinds are interrelated. Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth. It can be applied to the study of logical systems in which case category theory is called “categorical doctrines” at the syntactic, proof-theoretic, and semantic levels. Category theory even leads to a different theoretical conception of set and, as such, to a possible alternative to the standard set theoretical foundation for mathematics. As such, it raises many issues about mathematical ontology and epistemology. Category theory thus affords philosophers and logicians much to use and reflect upon.

Fun example to say “justified true belief” is not enuff

There is a widely accepted definition of “knowledge” as “justified true belief” .

A guy named Fred was in an accident in September 1955, and arrived at the hospital alive but unconscious. They managed to keep him alive for many years, though still unconscious. He finally woke up in July of 1969.

The medical team wanted to take it slow in introducing him to his now-aging family and generally orienting him to the modern world. In particular they did not tell him the current date nor explain how long he had been out.

“But there is one item of current news I’m very concerned about! How is Ike doing? Did he survive that heart attack??” [In point of fact, President Dwight Eisenhower had a heart attack in September 1955, and was thought to be in danger of his life, at the time Fred had his accident. But he rallied, and lived on, until March of 1969.]

Fred’s readjustment team went ahead and answered this emphatic query. “Oh Fred, I’m sorry to tell you, President Eisenhower is deceased.”

“You mean that sonovabitch Nixon is President?!” Fred exclaimed.

• So, we see that Fred believes Nixon is President
• That is a true belief, as Nixon was in fact elected in November 1968 and took office in January 1969.
• Fred’s belief that Nixon is President is justified by (a) his memory of the news of Ike’s heart attack; (b) the information he has just received, that Ike is dead; (c) his knowledge of the process of Presidential succession; and (d) his (formally) valid reasoning from a,b,c to the conclusion that Nixon is President.

But do we really want to accept that Fred knows that Nixon is President? It’s pretty shaky!