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For the simpler case of 1-categories, consider a 1-category $C$ and an orthogonal factorization system $(L,R)$ on $C$. Let $C_L$ and $C_R$ denote the wide subcategories of $C$ corresponding to the collections $L$ and $R$ of morphisms, respectively. Let $F_L\colon C_L \to D$ and $F_R\colon C_R \to D$ be functors between 1-categories. These functors extend to a single functor $F\colon C \to D$ (i.e., $\left.F\right|_{C_L}=F_L$ and $\left.F\right|_{C_R}=F_R$ on the nose) if and only if the following conditions are both satisfied:

  1. $\left.F_L\right|_{C_L\cap C_R} = \left.F_R\right|_{C_L\cap C_R}$ (on the nose).
  2. If $f\circ g=u\circ v$ with $f,v\in L$ and $g,u\in R$, then $F_L(f)\circ F_R(g) = F_R(u)\circ F_L(v)$.

Additionally, the extension $F$ is unique if it exists.

I'd like to extend this to the $\infty$-categorical case. Let $C\in\mathbf{sSet}$ be a quasicategory and let $(S_L,S_R)$ be an orthogonal factorization system on $C$, in the sense of Definition 5.2.8.8 in Lurie's Higher Topos Theory or Definition 04PD (9.1.9.1 at the time of this question) in Lurie's Kerodon. Let $C_L$ and $C_R$ be the simplicial subsets of $C$ consisting of simplices whose 1-faces are in $S_L$ and $S_R$, respectively. Let $D \in \mathbf{sSet}$ be another quasicategory and $F_L\colon C_L\to D$ and $F_R\colon C_R \to D$ be functors, i.e., simplicial maps, satisfying $\left.F_L\right|_{C_L\cap C_R} = \left.F_R\right|_{C_L\cap C_R}$ on the nose. I'd like to find a common extension, $F\colon C \to D$, of $F_L$ and $F_R$.

My first question is: what would be the $\infty$-version of the following commutation condition? $$ f\circ g=u\circ v,\, f,v\in L,\, g,u\in R \implies F_L(f)\circ F_R(g) = F_R(u)\circ F_L(v) $$ In $(\infty,1)$-case, I assume that it should look like a structure, rather than a property, which it is in the case of 1-categories; however, I am unable to devise the correct formulation. As my second questionon, if a proof or a reference for this assertion is available, it would be quite helpful. Thank you in advance for any assistance.

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    $\begingroup$ Thinking of categories as monads in spans, the kind of structure that connects $C_L$ and $C_R$ is exactly a distributive law of one monads over another ncatlab.org/nlab/show/distributive+law. The condition you are writing is the compatibility of $F$ with this distributive law. In the higher categorical case this involve a fully coherent higher structure though, it isn't going something very easy to check... $\endgroup$
    – Simon Henry
    Commented Feb 8 at 19:13
  • $\begingroup$ @SimonHenry Thanks. If $C$ is the nerve of Reedy 1-category, and $S_L=C_{-}, S_R=C_{+}$, then does this change anything? $\endgroup$
    – gksato
    Commented Feb 9 at 4:23
  • $\begingroup$ I don't know. At first sight I would say no, but I wouldn't rule it out completely given how presheaf on Reedy categories can be defined inductively. $\endgroup$
    – Simon Henry
    Commented Feb 9 at 12:28

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