Church in the West
The Church in the West is what Decius calls the Church of Rome, lead by the Pope, in The Apostle Documents in Rise of the Tomb Raider.
It was the Church in the West that sent the Order of Trinity around 977 AD to put an end to the heretic teachings of the Undying Prophet in the Byzantine Empire and - if possible - to retrieve the Divine Source. The Church possibly excommunicated the Prophet first, as Lara find a Papal Bullae (Tokens of Faith set of Relics) during her time in Siberia.
It is not exactly clear what the relationship between the Church and the Order of Trinity is. It can be assumed that the Church has at least some influence on the Order. On the other hand it seems that the modern times Trinity does not necessarily ask permission before mounting a campaign, although some connection is insinuated in the final credits / cut scene of Rise of the Tomb Raider through the business card of Kennard Montez, a member of the Vatican.