If someone is a danger to themselves or others do the police have lawful authority to detain the person?

In some circumstances the person can be detained. He can be put on a psych hold (called a 5150 in California) for up to 72 hours. If the person has brandished a weapon or otherwise violated the peace he could be arrested. In most cases, though, unless actual harm has occurred or appears certain to occur, police rarely intervene.

Supplemental Information: Many states (most?/all?) have statutes covering Involuntary Emergency Commitments for Mental Observation. They are known by various names and Acts in various jurisdictions. Their use is limited only to certain classes of specifically authorized laypersons (e.g.: law enforcement/paramedics/etc) to involuntarily commit persons PERSONALLY observed by them who clearly appear to be an imminent threat to themselves or others due to apparent mental deficiency. These commitments are made to a hospital or mental facility where the involuntarily committed person is observed by medical professionals for a statute-limited period of time before rendering a medical/psychiatric opinion as to the mental condition of the admittee. Every one of these involuntary commitments must be accompanied by a signed affidavit of the admitting official setting forth the facts of the situation and what THEY observed. When done by law enforcement officers, these are NOT considered "arrests."