category theory

# Contents

## Idea

An isofibration is, roughly speaking, a functor $F: E \rightarrow B$ between categories such that every isomorphism $f$ in $B$ can be ‘lifted’ to an isomorphism in $E$. Here ‘lifted’ means that there is an isomorphism $g$ in $E$ such that $F(g) = f$.

Alternatively, an isofibration is the analogue of a Hurewicz fibration for the interval object in $\mathsf{Cat}$, the category of categories, given by the free-standing isomorphism. That this definition is equivalent to the former one will be established below.

It is the second of these definitions that often generalises better to higher categories.

## Definition

###### Definition

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

###### Remark

Equivalently: for any commutative diagram

in $\mathsf{Cat}$, where $I$ is the free-standing isomorphism, there is a functor $l: I \rightarrow E$ such that the following diagram in $\mathsf{Cat}$ commutes.

###### Remark

We can picture this as follows.

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

###### Remark

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

###### Remark

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.

## Equivalent definition

The following may at first seem a little surprising. It says that isofibrations have in fact a stronger lifting property, namely the analogue of that of a Hurewicz fibration with respect to the interval object in $\mathsf{Cat}$ given by the interval groupoid. This stronger lifting property is more conceptually fundamental with regard to finding the correct generalisation of isofibrations to higher categories, where ‘correct’ refers for instance to defining the fibrations of a model structure.

###### Proposition

Let $p: E \rightarrow B$ be a functor between categories. Then $p$ is an isofibration if and only if for every commutative diagram

in $\mathsf{Cat}$, where $I$ is the free-standing isomorphism, there is a functor $l: X \times I \rightarrow E$ such that the following diagram in $\mathsf{Cat}$ commutes.

###### Proof

The “only if” direction is immediate. Let us demonstrate that the “if” direction holds by constructing $l$. To this end, for any object $x$ of $X$, let $l_{x}$ be the unique functor such that the following diagram in $\mathsf{Cat}$ commutes, which exists since $p$ is an isofibration.

Here $\phi_{x}$ is the functor representing the isomorphism $g(x, i)$ of $B$, where $i$ is the arrow $0 \rightarrow 1$ of $I$.

• For any object $x$ of $X$, we define $l(0)$ to be $f(x)$, define $l(x, i)$ to be $l_{x}(i)$, define $l\left(x,i^{-1}\right)$ to be $l_{x}\left(x^{-1}\right)$, and define $l(1)$ to be $l_{x}(1)$.
• For any arrow $r: x \rightarrow x'$ of $X$, we define $l(r,0)$ to be $f(r)$, and define $l(r, 1)$ to be $l_{x'}(i) \circ f(r) \circ l_{x}\left(i^{-1}\right)$.

It is immediately checked that $l$ is well-defined, is indeed a functor, and fits into the required commutative diagram.

## 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, sometimes known as Lack fibrations 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