When Mr. Snippy was captured by the Arbitrator, he was contained within a stasis field together with the Biomatrix and the mug. Since the sniper did not want to witness the slow destruction of Earth by the black hole the Arbitrator created, he requested the Biomass to create a memory for him, he could escape to instead. Taking his wish to get as far as possible away from the depressing event a little too literally, the Biomatrix recreated not one of Snippy's own memories, but rather genetic memories from his ancestors for him. While Biomatrix had promised not to interfere with the events unfolding for the sniper, it broke its promise on the discovery of an anomaly in Snippy's genetic memory.
The memories are not 100% historically accurate, neither are they real time-travel. So nothing Snippy will do in those memories will change events, that already have happened. The concept is somewhat similar to using the Animus in Assassin's Creed.
While the Biomatrix is keeping up the illusion for Snippy, it seems as well to have an eye on the events unfolding on the Earth surface, keeping the sniper updated by imbedding 'nightmares' into the memories.
So far Snippy has seen memories of an ancestor and one who appears to be a descendent of his: