Contents

category theory

# Contents

## Definition

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.”

Notice that this definition of isofibration violates the 1-categorical principle of equivalence where it demands that $p(\psi)=\phi$ (which includes the requirement that $p(e') = b$): this condition is not invariant under equivalence of categories. If one changed the definition to involve just an isomorphism $p(\psi)\cong\phi$, then of course, any functor would qualify. But the point of isofibrations is rather to help present the (2,1)-category of categories/groupoids in terms of 1-categorical data. For more on this see below at As fibrations in canonical model structures.

## Properties

### General

Isofibrations have a number of good properties. For example, any strict pullback of an isofibration is also a weak pullback. (This is also explained by the role of isofibrations as the fibrations in the canonical model structures, see below.)

Any Grothendieck fibration or opfibration is an isofibration, but not conversely (unless $B$ is a groupoid).

### As fibrations in canonical model structures

The isofibrations are the fibrations in the canonical model structure on categories and the canonical model structure on groupoids. 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.

For groupoids the definition appears (called “star surjectivity” there) on p. 105 (3 of 30) in