Is there a formal CS definition of VCS and file versions?
You're thinking of a tweet by Isaac Wolkerstorfer (‏@agnoster):

git gets easier once you get the basic idea that branches are homeomorphic endofunctors mapping submanifolds of a Hilbert space.

Unfortunately, it's a joke. As the author wrote on Quora:

It was intended as firmly tongue-in-cheek. I actually love git, and I think its complexity is greatly overblown. At the same time, I'm sympathetic to the fact that advice from git gurus to novices can end up sounding like inscrutable gibberish.
It's not intended to have any deeper meaning. Attempts to analyze it in this fashion should be futile, but due to a bug in reality, you can actually make any sufficiently hand-wavy statement fit if you try hard enough.

This has been discussed on Programmers Stack Exchange and on Mathematics Stack Exchange.

Joke aside, there has been work of formalizing version control. One effort that allies theory and practice is the work on patch theory by David Roundy on Darcs. The main objective of the theory is to model merging and, in particular, conflict resolution. The Darcs wiki has an introduction to the theory and a few pointers as well as a bibliography (unmaintained so outdated if you want a recent view on the subject, but it does list a 2009 survey paper by Petr Baudiš) and a list of talks (which includes more recent material). There's also a wikibook. One seminal paper is A Principled Approach to Version Control by Andres Löh, Wouter Swierstra and Daan Leijen3.
Patch theory does lead to a categorical model, which has been more recently explored in A Categorical Theory of Patches by Samuel Mimram and Cinzia Di Giusto and Homotopical Patch Theory by Carlo Angiuli, Ed Morehouse, Daniel R. Licata and Robert Harper. In Mimram and Di Giusto's work, the model has files as objects and patches as morphisms. I think that makes merging a branch a functor — an endofunctor if you're working in a single repository. “Homeomorphic endofunctor” doesn't make sense to me. With homotopy theory involved, submanifolds of a Hilbert space might not be so far out...