nLab equifibration

Idea

The notion referred to as equiv-fibrations [Lack 2002] or equifibrations [Campbell 2020, Def. 4.1] (both meant to be short fort equivalence-fibrations) or Lack fibrations [Moser, Sarazola & Verdugo 2022] is a categorification of that of isofibrations, from category theory to 2-category theory. Roughly speaking, a 2-functor $p \colon E \rightarrow B$ between 2-categories is an equifibration if equivalences in $B$ can be ‘lifted’ to equivalences in $E$. However, there are some subtleties to the precise definition, discussed below.

Beware that there is unrelated terminology of equifibered natural transformations.

Explicit definition

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

Definition

A equifibration is a 2-functor $p \colon E \rightarrow B$ between 2-categories such that, for every object $e$ of $E$, and every 1-morphism $f \colon p(e) \rightarrow b$ 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-morphism $h \colon e \rightarrow e'$ of $E$ and every 2-isomorphism $\phi \colon f \rightarrow p(h)$ in $B$, there is a 2-isomorphism $\psi \colon g \rightarrow h$ 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-morphismd

in $\mathsf{2-Cat}$, the 2-category of 2-categories, there is a functor $l \colon Q \rightarrow E$ such that the following diagram of 1-morphisns 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-morphism 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 original article, speaking of equiv-fibrations:

• Stephen Lack, A Quillen model structure for 2-categories, K-Theory 26 2 (2002) 171-205 [ author’s webpage, zbmath:3A1017.18005]

Beware that Lack 2002 contains an error, not pertaining directly to the definition of a equifibration itself, but to obtaining a model category structure on the category of strict 2-categories with this definition (cf. canonical model structure on 2-categories). This was fixed in the following paper Lack 2004 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 3 (2004) 185-197 [Zentralblatt review author’s webpage]

In Lack 2004, equifibrations were unnamed (and simply referred to as fibrations).

The term equifibrations is used in:

• Alexander Campbell. A homotopy coherent cellular nerve for bicategories, Advances in Mathematics 368 (2020) 107138 [arXiv:1907.01999, doi;10.1016/j.aim.2020.107138]

The term Lack fibrations is used in:

• Lyne Moser, Maru Sarazola, Paulo Verdugo, A 2Cat-inspired model structure for double categories, Cahiers LXIII-2 63 2 (2022) 184-236 [arXiv:2004.14233]