The Cathach is a psalter, a collection of the Psalms from the Bible. The original was at Movilla Abbey, in the possession of Saint Finnian of Movilla. A visiting Columba made a copy of it miraculously in a single night, in order to have his own, but Finnian objected to this.
The question arose: was the copy owned by Finnian because he possessed the original, or by Columba because he made it? The High King of Ireland, King Diarmait Mac Cerbaill, famously proclaimed "To every cow belongs her calf, therefore to every book belongs its copy." Then (according to this version of the story), Columba raised an army to fight for his right to keep the copy. It is far likelier that the battle was a dynastic conflict because of the way Diarmait had assumed the kingship after his predecessor's death.
Another theory for the start of the battle is the violation of the rules of sanctuary when a man under Columba's protection was forcibly taken and executed, whereupon Columba raised an army against Diarmait. Columba was exiled for his actions and required to create as many souls for Christ as had been killed in the battle (3000). There is no evidence that his missionary work after that time was the result of exile rather than a desire to spread Christianity.
But as to the battle itself: it was in northwest Ireland in what is now County Sligo. You can visit the site of the "Cooldrumman Battlefield" at the foot of Ben Bulben. Modern scholars point out that the earliest references to the battle do not mention a book at all.
Was the Cathach so important that a battle could arise over it losing its uniqueness and its place on Movilla Abbey. To answer that, we need to take a closer look at the Cathach itself, and that's what we'll do next.