nLab equifibration



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:EBp \colon E \rightarrow B between 2-categories is an equifibration if equivalences in BB can be ‘lifted’ to equivalences in EE. 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.


A equifibration is a 2-functor p:EBp \colon E \rightarrow B between 2-categories such that, for every object ee of EE, and every 1-morphism f:p(e)bf \colon 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-morphism h:eeh \colon e \rightarrow e' of EE and every 2-isomorphism ϕ:fp(h)\phi \colon f \rightarrow p(h) in BB, there is a 2-isomorphism ψ:gh\psi \colon g \rightarrow h in EE such that p(ψ)=ϕp(\psi) = \phi.


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

in 2 Cat \mathsf{2-Cat} , the 2-category of 2-categories, there is a functor l:QEl \colon Q \rightarrow E such that the following diagram of 1-morphisns 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 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.


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)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'}.


The original article, speaking of equiv-fibrations:

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:

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

The term equifibrations is used in:

The term Lack fibrations is used in:

Last revised on March 10, 2024 at 02:48:23. See the history of this page for a list of all contributions to it.