What is Category theory: Definition and 57 Discussions
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.
Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself.
Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations from 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.
Category theory has practical applications in programming language theory, for example the usage of monads in functional programming. It may also be used as an axiomatic foundation for mathematics, as an alternative to set theory and other proposed foundations.
Kea tells me that a good first book to read to learn category theory is
Sets for Mathematics $45
F. William Lawvere and Robert Rosebrugh
Advanced undergraduate or beginning graduate students need a unified foundation for their study of mathematics. For the first time in a text, this book...
Im a mathematical physicist, and lately I keep reading papers that throw this jargon around, and its beginning to bother me that I don't know anything about it, it feels like a gap in my knowledge.
I was trained as a mathematician in undergrad so of course it is somewhat familiar to me...
I was wondering how set theory could be categorized. I've heard that category theory can serve as an alternate foundation for math. But a fundamental symbol in the language is ∈. How is this done with arrows?
I was trying to figure out how arrows might be elements of a set (that isn't not...
Alejandro recently conducted a poll on Category Theory
and the response was remarkably positive
out of 7 respondents to the question
"Do you know what a Category is?"
there were 6 who said they could quote the definition from memory.
the poll was in the "Isham New Quantization" thread...