# Coproducts of weak equivalences

In a model category $\mathcal{C}$ admitting a forgetful functor to simplicial sets, is the coproduct of weak equivalences a weak equivalence? Say even just coproducts indexed by $\mathbf{N}$. A reference?

For example, the model category of simplicial bi-modules over a commutative rings with fibrations and weak equivalences defined to be those that are fibrations/weak equivalences of underlying simplicial sets, satisfies this, because homology of chain complexes commutes with coproducts.

However, just as an example, what about simplicial commutative monoids with analogous model structure, for instance?

• For simplicial commutative rings coproducts are given by tensor products, and one can simply take two ordinary rings A and B such that Tor^1(A,B)≠0 and consider two cofibrant resolutions A'→A and B'→B, whose tensor product cannot be a weak equivalence. – Dmitri Pavlov Oct 1 '17 at 7:23

Dmitri's comment shows that the answer to your question is no. However, if you are looking at weak equivalences between cofibrant objects, then coproducts are again weak equivalences. See Lemma 4.7 in my paper on commutative monoids in general model categories: https://arxiv.org/pdf/1403.6759.pdf

• If the objects are cofibrant, isn't this fact standard? It follows from Ken Brown's lemma, for example. – Tim Campion Oct 3 '17 at 1:37
• Yeah, the proof in my paper boils Ken Brown's lemma IIRC. But I remember thinking it was a pain to apply directly. I mean, what's the functor? From a bunch of copies of the arrow category to one copy? But then, when you talk about cofibrant objects, it's more than just source and target being cofibrant. So I just wrote out the argument to be airtight. But yes, I knew it had to be true because of Ken Brown's lemma. – David White Oct 3 '17 at 11:52
• Since weak equivalences are closed under composition, it suffices to show that if $w: A \to A'$ is a weak equivalence, then $w + 1_B: A + B \to A' + B$ is a weak equivalence for $A,B$ cofibrant. This follows from Ken Brown's lemma using cobase-change along $\emptyset \to B$. At any rate, the fact that pushout along a cofibration preserves weak equivalences between cofibrant objects is itself a standard fact. – Tim Campion Oct 3 '17 at 16:23