Orthogonal factorization system in a model category Are there any model category $\langle\mathcal{M},\mathfrak{C},\mathfrak{F},\mathfrak{W}\rangle$ whose factorization systems $\langle\mathfrak{C}\cap\mathfrak{W},\mathfrak{F}\rangle$ and $\langle\mathfrak{C},\mathfrak{W}\cap\mathfrak{F}\rangle$ are both orthogonal, i.e., solutions to lifting problems are unique? In particular, does model structure on the category of chain complexes have such property?
 A: There are no "interesting" model structures where the factorisation systems have unique lifting. More precisely:

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*If trivial cofibrations are left orthogonal to fibrations, then trivial cofibrations between fibrant objects are isomorphisms, and every morphism between fibrant objects is a fibration.

*If trivial fibrations are right orthogonal to cofibrations, then trivial fibrations between cofibrant objects are isomorphisms, and every morphism between cofibrant objects is a cofibration.

*If both orthogonality conditions are satisfied, then weak equivalences between fibrant–cofibrant objects are isomorphisms, and every morphism between fibrant–cofibrant objects is both a fibration and a cofibration.

But if weak equivalences between fibrant–cofibrant objects are isomorphisms, then the higher homotopical structure in the model category must be trivial – so the model structure is not "interesting".
Let me prove claim 1. Suppose trivial cofibrations are left orthogonal to fibrations. If $f : X \to Y$ is a trivial cofibration between fibrant objects, then there is a morphism $g : Y \to X$ such that $g \circ f = \textrm{id}_X$; but then $f \circ g \circ f = f$ and $Y$ is fibrant, so the unique lifting property implies $f \circ g = \textrm{id}_Y$, as required. Thus, if $f : X \to Y$ is any morphism between fibrant objects, then it can be factorised as an isomorphism followed by a fibration – which means $f : X \to Y$ itself is already a fibration.
Claim 2 is formally dual to claim 1. Since every weak equivalence can be factorised as a trivial cofibration followed by a trivial fibration, claim 3 follows.
Let me also remark that in the situation of claim 1, the full subcategory of fibrant objects is a reflective subcategory, with the fibrant replacement functor being the reflector. Thus the full subcategory of fibrant objects has limits and colimits, so it is a model category with the same weak equivalences and the special property that every morphism is a fibration.
