nLab Lack fibration

Idea

A Lack fibration or equiv-fibration is a categorification of the notion of an isofibration to the setting of 2-categories. Roughly speaking, a functor $p:E \rightarrow B$ between 2-categories is a Lack fibration if equivalences in $B$ can be ‘lifted’ to equivalences in $E$. However, there are some subtleties to the precise definition, as will be discussed below.

Explicit definition

The following works for both weak and strict 2-categories and weak and strict 2-functors.

Definition

A Lack fibration is a functor $p: E \rightarrow B$ between 2-categories such that, for every object $e$ of $E$, and every 1-arrow $f: p(e) \rightarrow b$ of $B$ which is an equivalence, the following hold.

1. There is an object $e'$ of $E$ and an equivalence $g: e \rightarrow e'$ in $E$ such that $p(g) = f$.
2. For every 1-arrow $h: e \rightarrow e'$ of $E$ and every 2-isomorphism $\phi: f \rightarrow p(h)$ in $B$, there is a 2-isomorphism $\psi: g \rightarrow h$ in $E$ such that $p(\psi) = \phi$.

Remark

Let $Q$ be the free-standing equivalence. Then condition 1. in Definition is equivalent to saying that for every (strictly!) commutative diagram of 1-arrows

in $\mathsf{2-Cat}$, the 2-category of 2-categories, there is a functor $l: Q \rightarrow E$ such that the following diagram of 1-arrows in $\mathsf{2-Cat}$ (strictly!) commutes.

The same is true if $Q$ is the free-standing adjoint equivalence, due to the fact that any equivalence can be improved to an adjoint equivalence.

Remark

Remark can be compared with the definition of an isofibration of 1-categories as expressed by a lifting condition: the condition is exactly the same, with the free-standing isomorphism replaced by the free-standing equivalence.

Remark

Let us explore the second condition in Definition a little. Note that any 1-arrow which is 2-isomorphic to an equivalence is itself an equivalence. Thus $p(h)$ is an equivalence, and condition 1. then ensures that $p(h)$ lifts to an equivalence $g'$ in $E$ such that $p\left(g'\right) = p(h)$. Condition 2. expresses that $g'$ must be 2-isomorphic to $h$. This implies in particular that $h$ is an equivalence.

Putting everything together, condition 2. is equivalent to: any 1-arrow of $E$ which maps to $f$ under $p$ up to 2-isomorphism is an equivalence, and all equivalences of $E$ which map to $f$ under $p$ up to 2-isomorphism are 2-isomorphic to one another, in such a way that, given 1-arrows $g$ and $g'$ of $E$, a 2-isomorphism $\phi_{g}: f \rightarrow p(g)$ in $B$, and a 2-isomorphism $\phi_{g'}: f \rightarrow p\left(g'\right)$ in $B$, the 2-isomorphism $\psi : g \rightarrow g'$ in $E$ has the property that $p(\psi) = \phi_{g}^{-1} \circ \phi_{g'}$.

References

The following was where Lack fibrations were introduced. It contains an error, not pertaining directly to the definition of a Lack fibration itself, but to obtaining a model structure on the category of strict 2-categories with this definition.

• Stephen Lack, A Quillen model structure for 2-categories, K-Theory 26, No. 2, 171-205 (2002). Zentralblatt review author’s webpage

The afore-mentioned error was fixed in the following paper by using the free-standing adjoint equivalence rather than the free-standing equivalence in the generating acyclic cofibrations.

• Stephen Lack, A Quillen model structure for bicategories, K-Theory 33, No. 3, 185-197 (2004). Zentralblatt review author’s webpage

The definition of a Lack fibration is recalled in Definition 3.2 of the following paper.

• L. Moser, M. Sarazola, P. Verdugo, A 2Cat-inspired model structure for double categories, 2020. arXiv:2004.14233