There was no police force in the modern sense actice in ancient Rome, such forces being a modern development by themselves. There were organisations with public tasks in the city:

In 13 BC Augustus formed the Cohortes Urbanae[1], who were stationed with but not part of the Pretorian guard. There were originally 10 of them, with initially 500 men per cohort. Under Vitelius, that number rose to 1000. Their main duty was to keep public order, e.g. at public gatherings but also occasionally to guard against petty criminals[2].

Then in AD 6, Augustus then formed the Cohortes Vigilum. While their main duties were that of a fire brigade, they were also used as watchmen[3]. The Cohortes were recruited from freedmen who were given full citizenship afteer six years of service and there was supposed to be one cohort of initially 500 and later of 1000 men per city district.

The Cohortes Urbanae disappeared with the disbandment of the Pretorian Guard after the Battle at the Milvian Bridge (312 AD)[4].

There is also no record of the cohorts to have been expected to investigate and make arrests for petty or even major crime. In fact, during the republic there was a taboo against exercising military command within the Pomerium. Protection against criminals was expected to be the private duty of the citizen.

Sources

[1] Freis, Helmut. Die cohortes urbanae. Köl, Böhlau Verlag, 1967 (Epigraphische Studien 2, Bonner Jahrbücher, Beihefte 21)

[2] Suetonius Tranquillo. Divus Augustus 43.

[3] Cassius Dio. Roman History, 55.24

[4] Van Dam, Raymond. Remembering Constantine at the Milvian Bridge. Cambridge, University Press, 2011.