Has anybody attempted to define the notion of a factorization system "in" a derivator?
Something on the lines of this: let $\mathbb{D}$ be a (strong) derivator. We define the orthogonality relation in $\mathbb{D}$ to be the following, if we denote
- $\square$ the category $[1]\times[1]$ (the "universal lifting problem"), and $\not\!\square$ the category $[3]$ (the "universal solution");
- $u\colon \square\to \not \!\square$ the obvious functor;
- $p_L, p_R\colon [1]\to \square$ the embeddings of the left and right side of the square (idem for $q_L,q_R\colon [1]\to \not\!\square$)
Given $f,g\in \mathbb{D}([1])$, and motivated by the fact that this result seems to be true when $\mathbb{D}$ is the represented (pre)derivator $J\mapsto{\cal C}^J$, we say that "$f\perp g$" iff $u^*\colon \mathbb{D}(\not \!\square)\to \mathbb{D}(\square)$ induces an equivalence between the subcategory $\mathbf{S}(f,g)$ of solutions $X\in \mathbb{D}(\not\!\square)$ such that $q_L^*X = f, q_R^* X = g$ and the subcategory $\mathbf{P}(f,g)$ of problems $Y\in \mathbb{D}(\square)$ such that $p_L^*Y = f, p_R^* Y = g$.
The request that every lifting problem has a solution seems to be an essential surjectivity request for $u^*|_{\mathbf{S}(f,g)}$; the orthogonality relation generates the obvious Galois connection between (full, seen as set of objects?) subcategories ${\cal K}\subseteq \mathbb{D}([1])$; a factorization system on $\mathbb D$ is, now, a pair of classes ${\cal E,M}\subseteq \mathbb{D}([1])$ such that $\mathcal E = {}^\perp\mathcal M$, $\mathcal M = \mathcal E^\perp$, and such that the functor $\Phi\colon \mathbb{D}([2])\to \mathbb D([1])$ is an equivalence, when restricted to the subcategory of "commutative triangles" $T$ such that $d_2^*T\in\cal E$, $d_0^*T\in\cal M$, where $d_i\colon [1]\to [2]$ are the obvious faces.
There are several reasons why this notion seems unsatisfying (it only uses a small piece of information provided by $\mathbb{D}$), but I don't want to bias you for the moment. Instead, a bit of speculation: since any sensible definition must re-enact the classical theory, I'm led to hope that this, or another better definition would entail that: $\def\E{\mathcal E} \def\M{\mathcal M} \def\D{\mathbb{D}} \def\op{\text{op}}$
${\cal E}\cap \cal M$ is the class of isomorphisms (but what is, precisely, an isomorphism in a derivator? An object $f\in\mathbb{D}([1])$ such that $\text{dia}_{[1]}(f)$ is an isomorphism in $\mathbb{D}(e)$? Or an object in the essential image of $\mathbb{D}(\{0\cong 1\})\to \mathbb{D}([1])$? Are these classes the same?)
The following conditions are equivalent for $f\in\mathbb{D}([1])$: (i) $f\perp f$ (ii) $f\perp \mathbb{D}([1])$ (iii) $\mathbb{D}([1])\perp f$; (iv) $f$ is an isomorphism (same question as above).
- If $(\E,\M)$ is a FS on $\D$, then $(\M^\op, \E^\op)$ is a FS on $\D^\op$.
- Let $\mathbf A$ be a bicomplete category; a FS on $\mathbf A$ induces a FS on $\D_\mathbf A$ (the represented prederivator $J\mapsto {\mathbf A}^J$).
- $\mathbf{M}$ a model category, define $J\mapsto \textsf{Ho}(\mathbf{M}^J) = \D_{\mathbf{M}}(J)$; then a Homotopy Factorization System $(\E, \M)$ (in the sense of Joyal) on $\mathbf{M}$, induces a FS on $\D_\mathbf{M}$.
- $\mathcal{C}$ an $\infty$-category; $J\mapsto \tau_1(\mathcal{C}^{NJ}) = \D_{\infty,\cal C}(J)$ is a derivator. If $(\E,\M)$ is a quasicategorical FS on $\cal C$, then $???$ is a FS on $\D_{\infty,\cal C}$.
...and many more other things. Have you ever heard of something similar? Do you have any advice on how to improve the definition?