nLab enhanced factorisation system

Definition

An enhanced factorisation system on a 2-category is an orthogonal factorisation system on a 2-category such that, for any invertible 2-cell $\alpha : t e \Rightarrow m s$

there is a unique pair of a 1-cell $r : B \to C$ and invertible 2-cell $\beta : t \Rightarrow m r$ such that $r e = s$ and $\beta e = \alpha$.

References

The condition to be an enhanced factorisation system was observed in the following, though the concept was not isolated:

• A. J. Power, A general coherence result, J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. doi:10.1016/0022-4049(89)90113-8 MR0985657

Enhanced factorisation systems were defined in

• Stephen Lack. Codescent objects and coherence. Journal of Pure and Applied Algebra 175 1-3 (2002) 223-241 [doi:10.1016/S0022-4049(02)00136-6]

attributed to Max Kelly.