On a recent afternoon in his office at Columbia University, Mr. Dinkins sat surrounded by plaques and photographs celebrating highlights from his career, a world apart from the arrows that once filled his days. At 90, the only African-American mayor in the city’s history, he has been a former mayor for one quarter of his life, three times as long as he held elective office. Across from his desk was a New York Newsday headline celebrating him as “Mayor Cool.”

“I sit here sometimes and I look and I reminisce,” Mr. Dinkins said, nodding toward a photograph of him with Harry Belafonte, a friend. Both men turned 90 this year. “He was the M.C. of my inauguration,” Mr. Dinkins said. “He was one of those who said to me: ‘You have to run. You must run.’ He insisted I run for mayor.”

Mr. Dinkins wore a red patterned bow tie and a blue double-breasted suit, filled out since he stopped playing tennis a few years ago. As he talked, his daughter called to ask whether he had seen a doctor about a nagging pain in his knee, and an alarm on his cellphone, programmed by his grandson, reminded him to take his several medications. These would have to wait; he left them at home.

He noted his unique place in New York lore. All mayors face criticism for problems that linger after they leave office; Mr. Dinkins’s critics focus on problems that quickly abated.

“The New York Times probably has an obit there for me now,” he said, raising a grievance he has aired before. He spoke with a courtly formality, quick with a set piece or a score to settle. “I always used to say, they’ll say, ‘David Dinkins, first black mayor of the City of New York,’ and the next sentence will be about Crown Heights,” the Brooklyn neighborhood where a four-day riot broke out on his watch, for which he was widely criticized.