Isofibration

A Displayed category is an isofibration if we can lift isomorphisms. Explicitly, a displayed category $\mathcal{E}\rightarrowtriangle\mathcal{B}$ is an isofibration if:

• For every object $e\rightarrowtriangle b$ and every iso $u:b\cong b^{\prime}$, there is an $e^{\prime}\rightarrowtriangle b^{\prime}$ and displayed iso $f:e\cong_{u}e^{\prime}$.

If the lifts are structure, then we would refer to this as a cloven isofibration. Conversely, if the lifts merely exist, then it is uncloven. By conventionn, I always use "isofibration" to refer to cloven isofibrations.