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Definition

Definition

Let $C$ be a category. A $k$-ary factorization system in $C$ is defined to be a (ordered) list $(L_1,R_1),\dots,(L_{k-1},R_{k-1})$ of (orthogonal) factorization systems such that ${R}_{i}\supset \supseteq {R}_{i+1}$ R_i\supset R_i\supseteq R_{i+1} (or equivalently $L_{i+1}\subseteq L_i$).

This is equivalent: Every $C$-morphism $f$ factors as $f=f_1;f_2;\dots;f_k$ with $f_i\in R_i\cap L_{i-1}$

Cassidy and Hébert and Kelly?, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)

Carboni and Janelidze? and Kelly? and Paré, “On localization and stabilization for factorization systems”, Appl. Categ. Structures 5 (1997), 1–58

Mike Shulman, internalizing the external - the joy of codiscreteness, blog

J. M. E. Hyland, 27.11.2012, Classical lambda calculus in modern dress, arXiv:1211.5762