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 between 2-categories is a Lack fibration if equivalences in can be ‘lifted’ to equivalences in . However, there are some subtleties to the precise definition, as will be discussed below.
The following works for both weak and strict 2-categories and weak and strict 2-functors.
A Lack fibration is a functor between 2-categories such that, for every object of , and every 1-arrow of which is an equivalence, the following hold.
Let be the free-standing equivalence. Then condition 1. in Definition is equivalent to saying that for every (strictly!) commutative diagram of 1-arrows
in , the 2-category of 2-categories, there is a functor such that the following diagram of 1-arrows in (strictly!) commutes.
The same is true if 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-arrow which is 2-isomorphic to an equivalence is itself an equivalence. Thus is an equivalence, and condition 1. then ensures that lifts to an equivalence in such that . Condition 2. expresses that must be 2-isomorphic to . This implies in particular that is an equivalence.
Putting everything together, condition 2. is equivalent to: any 1-arrow of which maps to under up to 2-isomorphism is an equivalence, and all equivalences of which map to under up to 2-isomorphism are 2-isomorphic to one another, in such a way that, given 1-arrows and of , a 2-isomorphism in , and a 2-isomorphism in , the 2-isomorphism in has the property that .
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.
Last revised on July 5, 2020 at 11:16:26. See the history of this page for a list of all contributions to it.