Is this map a cofibration in a model category?

Question

Consider the map $$i\colon A\sqcup A\to A$$ where $i=(Id_{A},Id_{A})$. Is this map a cofibration? Actually, I know that isomorphisms are cofibrations and pushouts of cofibrations are cofibrations. But I do not know what tools to use to prove that this map is a cofibration (if it is).

Answer 1

No, this is false. Consider the standard model structure on simplicial sets. The cofibrations are precisely the monomorphisms in this structure. However, unless $A$ is the empty simplicial set, your $i$ is never a monomorphism.