Univalence and completeness of Segal objects

On the ∞$\infty$‐topos semantics of homotopy type theory

“…The condition amounts to the following property: given any pair of pullback squares between small fibrations as below there exists an extension $$k$$ of $$h$$ along $$i$$ factoring the back pullback square as a composite of pullbacks. This has been referred to as strong gluing by Sterling and Angiuli [51], the stratification property by [50, 2.3.3], and acyclicity by Shulman [45, 47]. A more recent name for this condition is realignment [18].…”

“…We define $$D_1$$ and $$v$$ by the pullback and leave the rest of the verification to the literature [24, 43, 45]. Remark Note that fibrancy of $$U$$ was not required to prove the equivalence extension property, and in fact follows from it by an argument due to Stenzel [50, 2.4.3]. We must construct a lift that is, we must extend a fibration as indicated: Factor the composite $$j \cdot p_1$$ into an acyclic cofibration followed by a fibration and pull the fibration back: By right properness, $$j$$ pulls back to a weak equivalence and thus the induced map to the pullback is an equivalence between fibrations over $$A$$.…”

What should a generic object be?

“…Comment 70. In the context of universes, the realignment or acyclicity condition has been referred to as "Axiom (2 )" by Shulman (2015), as "strictification" by Orton and Pitts (2016), as "stratification" by Stenzel (2019), as "alignment" by Awodey (2021), and as "strict gluing" by Sterling and Angiuli (2021). See also Riehl (2022) for further discussion.…”

“…as "stratification" by Stenzel (2019), as "alignment" by Awodey (2021), and as "strict gluing" by Sterling and Angiuli (2021). See also Riehl (2022) for further discussion.…”