What do we want a foundation to do? Comparing set-theoretic, category-theoretic, and univalent approaches

http://www.socsci.uci.edu/%7Epjmaddy/bio/What%20do%20we%20want%20-%20final
(also published in Foundations of mathematics : logic at Harvard : essays in honor of W. Hugh Woodin’s 60th birthday
9781470440794, 1470440792)