## Definition +-- {: .num_defn} ###### 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 $R_i\supseteq R_{i+1}$ (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 $f_i\in R_i\cap L_{i-1}$ =-- ## Reflective factorization system [[reflective factorization system]] Relation of reflective subcategories and reflective subfibrations ## Cooperads ## Modalities Let $f:C\to D$ be a morphism in a (higher) category $\mathcal{C}$. The *$n$-image /Postnikov factorization (niP) of $f$* $$f=(C\simeq im_\infty(f)\to\dots\to im_n (f)\to\im_{n-1}(f)\to\dots\to im_1(f)\to im_0\simeq D)$$ is defined by (...) Let $\sharp$ be a (...) monad on $\mathcal{C}$. We consider the niP of the unit $\eta:C\to \sharp C$ which we denote by $$\eta=(C\simeq \sharp_\infty C\to \dots\to \sharp_n C\to \sharp_{n-1} C\to\dots\to \sharp_0\simeq \sharp C)$$ ## Examples ### Factorization systems in cohesive toposes [[pi-factorization systems]] ## References * Cassidy and Hébert and [[Max Kelly|Kelly]], "Reflective subcategories, localizations, and factorization systems". *J. Austral. Math Soc. (Series A)* 38 (1985), 287--329 ([pdf](http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_38_03%2FS1446788700023624a.pdf&code=5796045be8904c5183c2e95bce65491e)) {#CHK} * Carboni and [[George Janelidze|Janelidze]] and [[Max Kelly|Kelly]] and Paré, "On localization and stabilization for factorization systems", *Appl. Categ. Structures* 5 (1997), 1--58 {#CJKP} * Mike Shulman, internalizing the external - the joy of codiscreteness, [blog](http://golem.ph.utexas.edu/category/2011/11/internalizing_the_external_or.html) * J. M. E. Hyland, 27.11.2012, Classical lambda calculus in modern dress, [ arXiv:1211.5762](http://arxiv.org/abs/1211.5762) * UF-IAS-2012, Modal type theory, [wiki](http://uf-ias-2012.wikispaces.com/Modal+type+theory) * higher modalities, Michael Shulman, [pdf](http://uf-ias-2012.wikispaces.com/file/view/modalitt.pdf) * Steve Awodey, Nicola Gambino, and Kristina Sojakova. Inductive types in homotopy type theory. To appear in LICS 2012; [arXiv:1201.3898](http://arxiv.org/abs/1201.3898), 2012. 1 * Ching, _Bar construction for topological operads_ ([pdf](http://dspace.mit.edu/bitstream/handle/1721.1/27881/61212201.pdf?sequence=1))