Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial localization $L(C^D,W^D)$. Or I can first simplicially localize to get $L(C,W)$ and then form the $(\infty,1)$-functor category $L(C,W)^D$.

If $(C,W)$ is a nice model category, so that $(C^D,W^D)$ has a model structure presenting $L(C,W)^D$, then we have $L(C^D,W^D) \simeq L(C,W)^D$. Karol gave an easy counterexample of $C,W,D$ (without a model structure) for which it fails, depending on the fact that in general, morphisms in $L(C,W)$ may require zigzags of arbitrarily long length in $C$. But what if we assume some structure on $C$ less than a model category, but sufficient to ensure that zigzags of bounded length suffice?

For concreteness, how about if we assume a 3-arrow calculus as defined in the DHKS book Homotopy limit functors on model categories and homotopical categories? Recall that this means we have subcategories $U,V$ of $C$ satisfying the "functorial left (right) Ore condition" respectively and such that every map in $W$ factors functorially as one in $U$ followed by one in $V$. This ensures that every morphism in $L(C,W)$ can be represented by a 3-term zigzag, like in a model category.

I'd also be interested in answers to similar questions, such as what if $W$ admits a left or right calculus of fractions.

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial localization $L(C^D,W^D)$. Or I can first simplicially localize to get $L(C,W)$ and then form the $(\infty,1)$-functor category $L(C,W)^D$.

If $(C,W)$ is a nice model category, so that $(C^D,W^D)$ has a model structure presenting $L(C,W)^D$, then we have $L(C^D,W^D) \simeq L(C,W)^D$. Karol gave an easy counterexample of $C,W,D$ (without a model structure) for which it fails, depending on the fact that in general, morphisms in $L(C,W)$ may require zigzags of arbitrarily long length in $C$. But what if we assume some structure on $C$ less than a model category, but sufficient to ensure that zigzags of bounded length suffice?

For concreteness, how about if we assume a 3-arrow calculus as defined in the DHKS book Homotopy limit functors on model categories and homotopical categories? Recall that this means we have subcategories $U,V$ of $C$ satisfying the "functorial left (right) Ore condition" respectively and such that every map in $W$ factors functorially as one in $U$ followed by one in $V$. This ensures that every morphism in $L(C,W)$ can be represented by a 3-term zigzag, like in a model category.

I'd also be interested in answers to similar questions, such as what if $W$ admits a left or right calculus of fractions.