Spahn
factorization system

Definition

Definition

Let CC be a category. A kk-ary factorization system in CC is defined to be a (ordered) list (L 1,R 1),,(L k1,R k1)(L_1,R_1),\dots,(L_{k-1},R_{k-1}) of (orthogonal) factorization systems such that R iR i+1R_i\supseteq R_{i+1} (or equivalently L i+1L iL_{i+1}\subseteq L_i).

This is equivalent: Every CC-morphism ff factors as f=f 1;f 2;;f kf=f_1;f_2;\dots;f_k with f iR iL i1f_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:CDf:C\to D be a morphism in a (higher) category 𝒞\mathcal{C}. The nn-image /Postnikov factorization (niP) of ff

f=(Cim (f)im n(f)im n1(f)im 1(f)im 0D)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 𝒞\mathcal{C}. We consider the niP of the unit η:CC\eta:C\to \sharp C which we denote by

η=(C C nC n1C 0C)\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