# When category theory and set theory meet

1. Dec 10, 2003

### MathematicalPhysicist

was there an attempt to unite between those two fields?

2. Dec 10, 2003

### Ambitwistor

I don't know what you mean by "unite", but category theory can be applied to set theory, just like it can be applied to many other fields of mathematics.

For example, the standard set theoretic definitions of injection (one-to-one) and surjection (onto) for a function $f:S\rightarrow T$:

Injection: a map f such that if f(s1) = f(s2), then s1=s2.

Surjection: a map f such that for every element t in the range, there is an element s in the domain such that f(s) = t.

Category-theoretic definitions, which refer only to functions between sets, rather than elements of sets:

Injection (monomorphism): a map f such that for any two maps $u,v:W\rightarrow S$ for any set W, $f\circ u = f\circ v$ implies u=v.

Surjection (epimorphism): a map f such that for any two maps $u,v:T\rightarrow W$ for any set W, $u\circ f = v\circ f$ implies u=v.

For a whole treatment of set theory from the perspective of categories, read the gentle Conceptual Mathematics: A first introduction to categories by Lawvere and Schanuel.

An application of category theory to finite sets, with possible connections to physics:

http://arXiv.org/abs/math.QA/0004133

3. Dec 10, 2003

### BigRedDot

I don't know a whole lot about category theory, but I do know that set theory is subsumed by category theory. (And what isn't? I can't think of a more abstract or overarching theory than category theory.) Basic Category Theory for Computer Scientists has this to say:

"The category Set has sets as objects and total functions between sets as arrows. Composition of arrows is set-theoretic function composition. Identity arrows are identity functions."