PC 830.1(a)(2) and (3) are key among the provisions:

830.1. (a) Any sheriff, undersheriff, or deputy sheriff, employed
in that capacity, of a county, any chief of police of a city or
chief, director, or chief executive officer of a consolidated
municipal public safety agency that performs police functions, any
police officer, employed in that capacity and appointed by the chief
of police or chief, director, or chief executive of a public safety
agency, of a city, any chief of police, or police officer of a
district, including police officers of the San Diego Unified Port
District Harbor Police, authorized by statute to maintain a police
department, any marshal or deputy marshal of a superior court or
county, any port warden or port police officer of the Harbor
Department of the City of Los Angeles, or any inspector or
investigator employed in that capacity in the office of a district
attorney, is a peace officer. The authority of these peace officers
extends to any place in the state, as follows:

(1) As to any public offense committed or which there is probable
cause to believe has been committed within the political subdivision
that employs the peace officer or in which the peace officer serves.

(2) Where the peace officer has the prior consent of the chief of
police or chief, director, or chief executive officer of a
consolidated municipal public safety agency, or person authorized by
him or her to give consent, if the place is within a city or of the
sheriff, or person authorized by him or her to give consent, if the
place is within a county.

(3) As to any public offense committed or which there is probable
cause to believe has been committed in the peace officer's presence,
and with respect to which there is immediate danger to person or
property, or of the escape of the perpetrator of the offense.