snCaesarea was a city on the coast of Palestine south of Mount Carmel (not Caesarea Philippi). It was known as “Caesarea by the sea” (BDAG499s.v.Καισάρεια 2). Largely Gentile, it was a center of Roman administration and the location of many of Herod the Great’s building projects (Josephus, Ant. 15.9.6 [15.331–341]).
snA centurion was a noncommissioned officer in the Roman army or one of the auxiliary territorial armies, commanding a centuria of (nominally) 100 men. The responsibilities of centurions were broadly similar to modern junior officers, but there was a wide gap in social status between them and officers, and relatively few were promoted beyond the rank of senior centurion. The Roman troops stationed in Judea were auxiliaries, who would normally be rewarded with Roman citizenship after 25 years of service. Some of the centurions may have served originally in the Roman legions (regular army) and thus gained their citizenship at enlistment. Others may have inherited it, like Paul.
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sn A cohort was a Roman military unit of about 600 soldiers, one-tenth of a legion (BDAG936s.v.σπεῖρα). The Italian Cohort has been identified as cohors II Italica which is known to have been stationed in Syria in a.d. 88.