An isofibration is a functor$p:E\to B$ such that for any object$e\in E$ and any isomorphism$\phi:p(e) \cong b$, there exists an isomorphism $\psi:e \cong e'$ such that $p(\psi)=\phi$.

$\array{
e &\stackrel{\exists \psi \in p^{-1}(\phi)}{\to} & \exists e'&&& E
\\
&&&&& \downarrow^p
\\
p(e) &\stackrel{\phi}{\to} & b &&& B
}
\,.$

If $p$ is a forgetful functor, then being an isofibration says that whatever stuff $p$ forgets can be “transported along isomorphisms.”

Properties

Isofibrations have a number of good properties. For example, any strict pullback of an isofibration is also a weak pullback. Any Grothendieck fibration or opfibration is an isofibration, but not conversely (unless $B$ is a groupoid).

The isofibrations are the fibrations in the canonical model structure on Cat. More generally, the fibrations in canonical model structures on various types of higher categories are usually some generalization of isofibrations. For example, the fibrations in the Lack model structure on 2-Cat have “equivalence lifting” and “local isomorphism lifting,” and the fibrations in the Joyal model structure for quasicategories have “equivalence lifting” at all levels.

Generalizing in another direction, internalized isofibrations are the fibrations in the 2-trivial model structure on any finitely complete and cocomplete strict 2-category.

This definition of isofibration is evil where it demands that $p(\psi)=\phi$; if it only demanded $p(\psi)\cong\phi$, of course, any functor would qualify.

Revised on January 28, 2012 09:34:29
by Mike Shulman
(71.136.231.206)