Cave currently lives and works in Chicago where he is a professor and Chair of the Fashion department at the Art Institute of Chicago. He has held solo exhibitions at the Denver Art Museum in 2013; Institute of Contemporary Art/Boston in 2014; St. Louis Art Museum in 2014–15; and Cranbrook Art Museum in 2015.
His one-person exhibition Until opened at the Massachusetts Museum of Contemporary Art in 2016, travelling to Carriageworks, Sydney and Crystal Bridges Museum of American Art, Bentonville in 2018. He received the Americans for the Arts 2014 Public Art Network Year in Review Award in 2014; the Joan Mitchell Foundation Award in 2008; and the Artadia and Joyce Awards both in 2006, among others.