Both. The Byzantine Empire had had military units dedicated to the protection of the Emperor and the capitol from the beginning, although the names of these units, their structure, function, and organization would change dramatically in different periods.
Some of these units were used as police – the Tzakones started out as marines and ended up as palace guards, the Vardariotai were basically riot police armed with whips and staves, the Skythikoi were mercenaries who also were drafted in as police at various times.
But there were also a regular city police force called the taxiōtai (as well as a night watch) who served under the urban prefect of Constantinople, who was in charge of the jails and the law courts.
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