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Stream: theory: category theory

Topic: lax fibrations?

Matteo Capucci (he/him) (Nov 27 2021 at 13:37):

Does anyone know if this following 'laxification' of the notion of 2-fibration is known?

Given a 2-functor $P : \mathcal E \to \mathcal B$, it is a 'lax 2-fibration' if

1. for each morphism $u: X \to PY$ in $\mathcal B$, there is a 2-cartesian morphism $\bar u : u^*(X) \to Y$ together with a 2-morphism $\varphi : P\bar u \Rightarrow u$,
2. for each 2-morphism $\alpha : u \Rightarrow v : X \to PY$ in $\mathcal B$, there is a 2-cartesian 2-morphism $\bar \alpha : f \Rightarrow g : u^*(X) \to Y$ such that $P\bar \alpha = \alpha$

The bold and italic part is what is different from the usual definition, where $\varphi$ is assumed to be always the identity:

Matteo Capucci (he/him) (Nov 27 2021 at 13:38):

Screenshot-from-2021-11-27-13-35-39.png