Mayor Bill de Blasio took the oath of office on Monday, a frigid New Year’s Day, to begin his second and last term, with broad promises for building a fairer city and with his sights already fixed beyond the five boroughs.

He was sworn in by the populist senator and leftist icon Bernie Sanders of Vermont, in an outdoor ceremony that was truncated because of the cold — a youth chorus and the city’s youth poet laureate were eliminated from the program, and Mr. de Blasio said that he shortened his remarks out of consideration for his shivering audience. The temperature was in the teens, under a powdery blue sky, with a sharp wind that picked up once the mayor began speaking.

Several hundred people, including other elected officials, watched the ceremony in the plaza in front of City Hall, although empty seats were scattered throughout — a contrast to four years ago when thousands packed Mr. de Blasio’s first swearing in (the weather then was warmer).

Aside from Mr. Sanders, the political star power was significantly reduced from Mr. de Blasio’s first inauguration. Then, he was sworn in by former President Bill Clinton and other attendees included former Secretary of State Hillary Clinton, Gov. Andrew M. Cuomo and the activist and performer Harry Belafonte. None of them were there on Monday.