# Inverses and Isomorphism

One big plus to this approach is that unifying ideas from all over math means we only have to prove theorems once. Here’s an example:

Let $\mathcal C$ be a category, and let $X$ and $Y$ be objects in $\mathcal C$. Further, let’s suppose we have a map $f:X\to Y$. If there is some $g:Y\to X$ such that $g\circ f=1_X$, then we say that $g$ is a left-inverse of $f$. Similarly, if $f\circ g=1_Y$, then $g$ is a right-inverse of $f$. If it’s both, then we just say that $g$ is an inverse of $f$, and that $X$ and $Y$ are isomorphic ($f$ and $g$ are isomorphisms).

So there’s a definition of two objects being isomorphic in an arbitrary category. Kinda cool. We can now say that two sets are isomorphic if there is a bijection between them. Two vector spaces are isomorphic if an invertible linear transformation takes one to the other. Two groups are isomorphic if there is a bijection between them that preserves multiplication and inversion. Of course, we knew all of this, but now we can say it more generally and in a uniform way.

#### Theorem: Inverses are unique:

Suppose $g$ and $g'$ are inverses of $f$. The proof is simply an exercise in associativity and inverses…

$g=1_X\circ g=(g'\circ f)\circ g=g'\circ(f\circ g)=g'\circ 1_Y=g'$.

$\square$

The nice thing about categories is that we are now done with proving uniqueness of inverses. We never have to do it again. If later, we’re working with some wacky mathematical object, and we see that two maps are inverses of each other… BAM! Uniqueness is immediate, because we’ve done it in full generality.