nLab equifibration

1. 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.

2. Explicit definition

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

Definition 2.1. 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 2.2. Let $Q$ be the free-standing equivalence. Then condition (1.) in Definition 2.1 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 2.3. Remark 2.2 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 2.4. Let us explore the second condition in Definition 2.1 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 $p(\psi) = \phi_{g}^{-1} \circ \phi_{g'}$.

3. 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]