Does the following categorial sum preserve weak equivalences? - MathOverflow

Does the following categorial sum preserve weak equivalences?

2012-10-02T06:40:16Z

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.

Answer by Karol Szumiło for Does the following categorial sum preserve weak equivalences?

2012-10-02T09:10:14Z

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.