We give an overview of the menagerie of different notions of cubical set. First, there are a number of inequivalent categories deserving of the name “category of cubical sets” (plain cubical sets, cubical sets with connections / symmetries, etc). Each one is a presheaf category on a small category which we call a cubical site. We will follow
Buccholtz and Morehouse in using Mauri’s theory of monoidal logic to systematically enumerate the possible choices of cubical site, and then discuss some more concrete ways of thinking about these various sites. We will also show that most of these sites are Eilenberg-Zilber categories, giving some combinatorial traction to their study.
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