(This is a series of articles on police as secondary characters in mystery novels. The other entries in the series are Columbo Ex-Machina, The Dismissive Policeman, and Policeman as Community Member.)
As I mentioned last time, Inspector Slack might be at odds with Miss Marple, might resent her, but he was always on the same side as she was. They had the same goal.
But there is a common police character who does not have the same goal as the protagonist. Sometimes the policeman is the actual antagonist of the story. This happens most often in stories that aren't actually mysteries, but rather straight crime stories. Columbo, Crime and Punishment, Les Miserables, The Fugitive. And for that matter, Raffles, Arsene Lupin, Zorro, The Scarlet Pimpernel and The Saint.
When your hero is outside the law - whether he's a criminal, or accused of a crime, or some shadowy vigilante - it's the job of the policeman to stop him. These stories are often games of cat and mouse, and the policeman is a foil for the protagonist. Columbo, and Porfiry Petrovich (of Crime and Punishment) are mild, ordinary, hard working characters facing off against arrogant criminals. In the movie version of The Fugitive, Richard Kimball is an amateur at being a fugitive, and he bumbles his way through by the skin of his teeth, while Marshal Sam Gerard pursues with menacingly super-competent precision. Gentleman burglars and swashbucklers tend to be chased down by working class plodders.
Sometimes such characters are downright villainous - as with Citizen Chauvelin, who is the nemesis of the Scarlet Pimpernel. Chauvelin seems to be a true believer in the revolution, but he keeps secrets from his superiors, and plays games to meet his own ambitions. Like all villains, he's really after power and self-aggrandizement. He's willing to murder men, women and children, and he's equally willing to release enemies of the revolution to further his own political ends. He's a blackmailer, and a liar. (And yet he still manages to be somewhat smarter and better than the evil men he works for - who can't seem to moderate their evil, even when it's in their own best interests. They often try to intimidate and blackmail Chauvelin, and he plays them for a bigger prize than he might otherwise get.)
In the mystery, however, the policeman is usually some sort of good guy. He may be like Sam Gerard or Javert from Les Miserables, and be frighteningly single-minded. Just think about the scene in The Fugitive when Gerard tracks Kimball into the drain pipes of the dam, and Kimball tells him "I did not kill my wife!" and Gerard says "I don't care!" It's not Gerard's job to judge. It's his job to catch, and he'll do it no matter what. However, when faced with the truth, he does do the right thing.
All nemesis characters have to be good at their jobs. Unlike the Inspector Slack type, the character has to be a good match in skills for the hero.
A good example of this would be Chief Inspector Teal of The Saint. Teal is in a difficult position, because he and The Saint are really on the same side when it comes to catching the bad guys. However, The Saint breaks the law, and Teal is equally out to get him. But it's nothing personal, and they often chat together and cooperate. Teal is not depicted as a fool. He is described in one book as so sharp, that if Scotland Yard's archives were to be destroyed in a fire, Teal could probably recreate them entirely from his own memory, including rough sketches of all the finger prints. Teal's main limits are that he is honest, and he must abide by the law... and also he's limited by the limits of his minions, who are really not up to the cleverness of The Saint.
I could go on for a while about this type of character, because I really like this model, and I'm working on a character who is a hybrid of this and some other kinds. But I think it's time to move on. I think I want to talk about the Plodder and the Mentor but I'm not quite sure of what I want to say yet.
(The other entries in the series are Columbo Ex-Machina, The Dismissive Policeman, and Policeman as Community Member.)