In the pilot episode of Sleepy Hollow, Abbie is identified as a lieutenant in the Westchester County PD, working under Sheriff August Corbin. Later, after Corbin's death, she introduces herself to Captain Frank Irving, who appears to be in command of the precinct that includes Sleepy Hollow. Irving does not seem immediately familiar with her until she identifies herself as Corbin's "partner". Lieutenant Mills subsequently begins taking orders directly from Irving.
I'm confused as to how the chain of command in that police department works. Sheriff is the highest-ranking official in a police department, usually an elected official. Lieutenant is fairly near the bottom, but still high enough that she should normally be a supervisory position, with one or more sergeant beneath her.
The fact that both the Sheriff and one of his Lieutenants still went on patrols, and did so together, implies that the police department must be tiny, but the Westchester County PD has dozens of officers. Even more strange, Corbin (as sheriff) would be responsible for all precincts in the county, making him Irving's boss, so the idea that Irving would not know how own boss's "partner" is also unlikely. Either Lieutenant Mills worked in Irving's precinct, in which case he obviously should know her, or she works in another precinct, in which case he isn't her superior and wouldn't routinely take his orders.
So where exactly are Lieutenant Mills, Captain Irving, and Sheriff Corbin supposed to fit into a chain of command?
For reference, the department on the show is clearly not drawing on real-life for its model. The real Westchester County Police Department has no sheriff. That position is held by the head of the county-wide Westchester County Department of Public Safety, while the Westchester County Police Department only covers those areas not part of another precinct. In particular, Sleepy Hollow has it's own PD. The highest ranking official in a single precinct is the police commissioner (whom we don't see).