What it says on the tin. Did the Roman Empire have any military or judicial units whose sole purpose was to investigate suspicious deaths, interview witnesses and present evidence of a crime? I'm talking about crimes that affected both the noble families and the peasantry.
You may want to read this very recent thread asking about the Praetorian Guard, which explains that among other duties they acted as a police force.
As specified in [this] (https://www.reddit.com/r/AskHistorians/comments/27kg1t/did_the_roman_army_have_special_units/ci1tfj4earlier) post, the urban cohort did exist as a dedicated police force in the city of Rome.
Not Roman but Athens had a force of 300 Scythian Archers who were essentially a police force and acted as crowd control at assemblies and festivals. http://www.stoa.org/projects/demos/article_scythian_archers?page=all&greekEncoding=