Gifford describes Stephen’s "Latin quarter hat" as “A soft or slouch hat associated with the art and student worlds of the Latin Quarter in Paris, as against the 'hard' hats (bowlers or derbies) then fashionable in Dublin.” Joyce wore such a hat in 1902, and mailed a photograph showing it to a friend.
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Gifford, Don and Seidman, Robert J. Ulysses Annotated: Notes for James Joyce's Ulysses. Berkeley: University of California Press, 2008. Print.