There was no permanent public investigative force in the city, period. Maintenance of public order and security was the duty of the various magistrates, particularly the aediles and praetors and (in capital cases) the tresviri capitales, as well as their auxiliary staff. By and large, however, public officials neither arrested criminals nor investigated crimes before they were brought to court--I address the question of arrest and investigation in this thread, although you'll have to go through the followup comments to get to it. Generally "criminal justice" as we might understand it was the duty of the private citizen, not the state, although in certain capital cases the state might behave differently. But it was usually the duty of the wronged party to bring suit or criminal accusation and of the prosecution to muster evidence for trial, not the state.

The cohortes urbanae should not be thought of as a police force or anything resembling one. They were a paramilitary force overseen by the city prefect, who under the Principate was charged with maintenance of public order. The city prefect had his own law courts, but these were not connected with the urban cohorts, which were strictly a paramilitary force for quelling public disturbances