At first glance, quotient categories might seem like a strange notion to define, but in actuality they turn out to be quite useful constructions. On one hand, they simplify the treatment of certain categories. On the other hand they arise naturally in geometry as a way of studying spaces which may not be distinguishable using more elementary algebraic methods. In this talk, we explore the theory of quotient categories and how this theory interacts with that of sheaves and vector bundles on the projective line. In particular, we will see how this theory can be used to prove Grothendieck's Splitting Theorem for vector bundles.