Barack Hussein Obama renewed his oath of office at midday Monday, ceremonially marking the beginning of another four years in the White House and embracing a progressive agenda centered on equality and opportunity.

Mr. Obama went out of his way to mention both gay rights and the need to address climate change, in a speech that seemed intended to define his version of modern liberalism, shortly after voters returned him to office for a second term.

“For now decisions are upon us, and we cannot afford delay. We cannot mistake absolutism for principle, or substitute spectacle for politics, or treat name-calling as reasoned debate,” Mr. Obama said. “We must act; we must act knowing that our work will be imperfect.”