Spahn factorization system

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 system

Pi-closure

References

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

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