In Dwyer and Kan's 1980 paper on "Simplicial Localizations of Categories", they prove the following result for binary categorial sums. For a set $O$, let $O$-${\mathsf{Cat}}$ be the following category. An object of $O$-${\mathsf{Cat}}$ is a small category with $O$ as its set of objects. An morphism of $O$-${\mathsf{Cat}}$ is a functor that restricts to the identity on objects. Let $sO$-${\mathsf{Cat}}$ be the category of simplicial objects in $O$-${\mathsf{Cat}}$. 


A map $A\to B$ in $sO$-${\mathsf{Cat}}$ is a *weak equivalence* if, for every two objects $X,Y\in O$, the induced map of simplicial sets ${\mathrm{Hom}}\_A(X,Y)\to {\mathrm{Hom}}\_B(X,Y)$ is a weak homotopy equivalence. Kan and Dwyer's result is that if two maps $A\to A'$ and $B\to B'$ in $sO$-${\mathsf{Cat}}$ are weak equivalences, then so is their categorial sum $A+B\to A'+B'$.


My question is whether this result extends to infinite categorial sums? Kan and Dwyer's approach is to show that if $A\to A'$ is a weak equivalence then $A+B\to A'+B$ is a weak equivalence. Hence their approach does not generalize to the infinite case. 



Does the following categorial sum preserve weak equivalences?
The general case can be concluded from the finite one as follows. Let $(A\_i \to B\_i \mid i \in I)$ be a family of weak equivalences between simplicial $O$-categories. I'm going to assume that $I = \mathbb{N}$, the general case can be handled similarly, but the notation would be a bit more tedious (you can well-order $I$ or consider the directed poset of finite subsets of $I$).


The coproduct $\coprod\_{i \in \mathbb{N}} A\_i$ can be written as a colimit of the sequence


$$A\_0 \to A\_0 \sqcup A\_1 \to A\_0 \sqcup A\_1 \sqcup A\_2 \to \ldots$$


which is computed hom-set-wise (since morphisms in the big coproduct can be written as finite composites of morphisms of $A\_i$s so each occurs at some stage in the sequence). Moreover, all the maps induced on hom-sets are injective i.e. cofibrations of simplicial sets. The transformation between this sequence and the corresponding one for $B\_i$s is a natural weak equivalence by the result you mentioned. Putting this all together you get that the induced map on each hom-set in the colimit is also a weak equivalence.


