nLab
Lack fibration

Contents

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:EBp:E \rightarrow B between 2-categories is a Lack fibration if equivalences in BB can be ‘lifted’ to equivalences in EE. 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:EBp: E \rightarrow B between 2-categories such that, for every object ee of EE, and every 1-arrow f:p(e)bf: p(e) \rightarrow b of BB which is an equivalence, the following hold.

  1. There is an object ee' of EE and an equivalence g:eeg: e \rightarrow e' in EE such that p(g)=fp(g) = f.
  2. For every 1-arrow h:eeh: e \rightarrow e' of EE and every 2-isomorphism ϕ:fp(h)\phi: f \rightarrow p(h) in BB, there is a 2-isomorphism ψ:gh\psi: g \rightarrow h in EE such that p(ψ)=ϕp(\psi) = \phi.

Remark

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

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

The same is true if QQ 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)p(h) is an equivalence, and condition 1. then ensures that p(h)p(h) lifts to an equivalence gg' in EE such that p(g)=p(h)p\left(g'\right) = p(h). Condition 2. expresses that gg' must be 2-isomorphic to hh. This implies in particular that hh is an equivalence.

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

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.

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

Last revised on July 5, 2020 at 07:16:26. See the history of this page for a list of all contributions to it.