Fibrations and Anodyne Maps
Table of Contents
1. Fibrations
Fibrations are special functors between \(\infty \)-categories. We recall all notions of fibrations and collect results about fibrations.
Let \(p \colon X \rightarrow Y \) be a morphism of simplicial sets. \(p \) is a (left, right, inner) fibration if \(p \) has the RLP with respect to all (left, right, inner) horn inclusions. \(p \) is a trivial fibration if \(p \) has the RLP with respect to all boundary inclusions.
Joyal fibration isofibration
Let \(f \colon \mathcal C \rightarrow \mathcal D \) be a functor of \(\infty \)-categories. \(p \) is a Joyal fibration if \(p \) is an inner fibration that has the RLP with respect to the map \(\Delta^0 \rightarrow J \).
Let \(f \colon \mathcal C \rightarrow \mathcal D \) be a functor of \(\infty \)-categories. \(p \) is an isofibration if \(p \) is an inner fibration and every lifting problem
\begin{equation*} \xymatrix{ \{0\} \ar[r] \ar[d] & \mathcal C \ar[d] \\ \Delta^1 \ar[r]^f \ar@{-->}[ur] & \mathcal D } \end{equation*}where \(f \) is an equivalence in \(\mathcal D \) has a solution that is an equivalence in \(\mathcal C \).
Left and right fibraitons between \(\infty \)-categories are conservative isofibrations.
Let \(p \colon \mathcal C \rightarrow \mathcal D \) and \(f \colon \mathcal D ' \rightarrow \mathcal D \) be a functors of \(\infty \)-categories. If \( p\) is a conservative inner fibraton, then the pullback \(f^*p \) of \(p \) along \(f \) is a conservative inner fibration. If \(p \) is an isofibration, then \(f^*p \) is an isofibration.
Let \(p \colon \mathcal C \rightarrow \mathcal D \) be a functor of \(\infty \)-categories. Then \(p \) is an inner fibration if and only if \(p \) is a Joyal fibration.
characterization of isofibrations (2.1.20, 2.1.21)