2026/07/20 by David Gepner, Hadrian Heine
#math.AT #math.CT
We study fibrations of higher categories from the perspective of oriented category theory, a framework which accounts for lax phenomena in higher category theory via systematic enrichment in the Gray tensor product. We give several equivalent characterizations of fibrations of (∞,∞)-categories and oriented categories, and show that categories of fibrations naturally organize to form oriented categories. We study the interaction between fibrations and oriented pullbacks and construct higher-categorical versions of free fibrations and universal fibrations. The latter give rise to Grothendieck constructions for fibrations of (∞,∞)-categories and oriented categories.