Coherence for factorization algebras (Q2768445)

It is well known that algebras for the 2-monad \(((-)^2,I,C)\) on \({\mathcal {CAT}}\) are related to factorization systems; here \((\mathcal K)^2\) denotes the category of arrows, \(I_\mathcal K:\mathcal K\to\mathcal K^2\) sends an object to its identity arrow and \(C\) is given by a composition. In fact, \textit{L. Coppey} [Diagrammes, Suppl. 3 (1980; Zbl 0497.18015)] showed that strict algebras are strict factorization systems and \textit{M. Korostenski} and \textit{W. Tholen} [J. Pure Appl. Algebra 85, No. 1, 57-72 (1993; Zbl 0778.18001)] proved that normal pseudo-algebras are equivalent to factorization systems (\(F:\mathcal K^2\to\mathcal K\) is normal if \(FI_{\mathcal K}=1_{\mathcal K}\)). (W. Tholen and the reviewer have recently shown that weak factorization systems are related to oplax algebras for \((-)^2\).) The authors prove a surprising result that a functor \(F:\mathcal K^2\to\mathcal K\) satisfying \(FI_{\mathcal K}=1_{\mathcal K}\)) admits a unique, normal pseudo-algebra structure iff there is a mere natural isomorphism \(\alpha:FF^2\to FC_{\mathcal K}\). Note that \(\alpha\) itself may fail to be an algebra structure.