Historical Fiction
Bishop Latour isn't a real person, but he's based on one: Jean Baptiste Lamy. This dude was really the first Archbishop of New Mexico, and totally built the Cathedral Basilica of Saint Francis of Assisi in Santa Fe.
He also had a dear friend named Joseph Machebeuf who is a template for Vaillant. Also, Cather throws in the character of Kit Carson without even changing his name: Carson was a real gunslinger who hung out in New Mexico during the mid-nineteeth century. He's represented pretty faithfully, from what we can tell:
Death Comes for the Archbishop was published in 1927, which means that Cather was writing about events from seventy years prior, when New Mexico first became part of the United States. We imagine she spent a bunch of time poring over history books, even while she was wandering around the New Mexico desert.
But all this research would have been fun for her, because Cather was riveted by the story of Lamy from the first time she set foot in Santa Fe:
''I never passed the life-size bronze of Archbishop Lamy which stands under a locust tree before the Cathedral in Santa Fe without wishing that I could learn more about a pioneer churchman,'' Miss Cather wrote later, noting that his countenance expressed ''something fearless and fine and very, very well-bred...'' (Source)