groupoids and localization in categories

Question

I've been learning homology (and hence Category Theory) and have been interested in groupoids for some time. I've gotten to localization in categories, and that has sparked the following question: Is the localization of a category a groupoid? So going further, isn't localization a functor from Cat (the category of categories) to groupoids (which can be seen as a sub-category of Cat)?

Answer 1

Let C be a category, and W a class of maps of C which satisfies the 2-3 property. This means that if two of the maps $f:X\to Y, g:Y\to Z, g\circ f$ is in W so is the third. Then other some circumstances, there is a notion of localization of W.

This is the purpose of the theory of closed models of Quillen: you need a class of cofibrations cof(C) and a class of fibrations fib(C) such that $(cof(C)\cap W,fib(C))$ and $(cof(C),fib(C)\cap W)$ are weak factorization systems. There exists a homotopy category which is a localization of W.

In general there are set theoretical problems which prevent to localize any class of maps in a given category.

Answer 2

No. When you localize you're inverting some of the morphisms, not all of the morphisms. There is a special localization where you try to invert all of the morphisms; this is (up to size issues) the left adjoint to the inclusion from groupoids into categories.