# Spahn factorization system (changes)

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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 $R_i\supseteq R_{i+1}$ (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

## 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 Pi-factorization systems 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
• 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

• UF-IAS-2012, Modal type theory, wiki

• higher modalities, Michael Shulman, pdf

• Steve Awodey, Nicola Gambino, and Kristina Sojakova. Inductive types in homotopy type theory. To appear in LICS 2012; arXiv:1201.3898, 2012. 1

• Ching, Bar construction for topological operads (pdf)

Last revised on December 2, 2012 at 18:28:30. See the history of this page for a list of all contributions to it.