The fight is similar to others being waged across the country, as cities weigh the best approach to address crime, homelessness and poverty. In Chicago, Brandon Johnson, the progressive Democratic candidate for mayor, won this month on a public safety message that went beyond policing, and focused on youth employment programs and mental health services. Mr. Adams and his top political adviser, Evan Thies, each discounted the notion that the mayoral election in Chicago was a sign that Democratic values had shifted to the left, and that New York may follow suit. When Lori Lightfoot, the outgoing mayor of Chicago, lost in the first round of voting last month, Mr. Adams said he did not view her loss as a warning sign for himself. “Both Mayor Adams and Mayor-elect Johnson ran campaigns that prioritized safety and justice, and were supported by large Black electorates,” Mr. Thies said in a statement.

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