Police forces as such did not exist in ancient world. They were created during 18th and 19th centuries. Ancient societies recognized the need for publicly appointed officials to carry out functions of of social regulation. In ancient Rome and cities of the Latin west these functions were performed by the aediles and their equivalents.
There were also magistrates appointed to deal with certain aspects of criminal activity. In Rome these were minor magistrates called tresviri capitales, who 'may also have exercised summary jurisdiction over slaves and humble citizens' (Oxford Classical Dictionary). These magistrates were only assisted by a small number of public slaves and had neither the authorityy nor the resources to act as a police force.
At Rome the lictors who attended the senior magistrates were only symbols of the state's authorityy to discipline and punish; they had no effective power to coerce. The authority of magistrates depended absolutely on the acceptance by the citizens of their political institutions and the men who operated them.
Generally the state was not getting involved into any investigation and prosecution of criminal activity, which were left to the citizens, their families and friends. Small-scale disturbances were resolved locally by neighbours and passers-by, who were expected to take sides and usually did so.
The state became involved only when violence had a political dimension or when it became a threat to the community as a whole. In such cases the authorities organized ordinary citizens who took u arms on behalf of the state. This happened in Rome in 186 BC at the time of the Bacchanalian affai (Livy 39. 16. 13). In the political crises of 121 and 100 BC the senators and knights armed themselves and their dependants in order to crush Gaius Sempronius Gracchus and L. Appuleius Saturninus.
The need to call upon the armed support of the citizens in a crisis was stated in Roman colonial charters (e.g., lex Coloniae...