Let $\mathbb{S}$ be an excellent model category (see https://ncatlab.org/nlab/show/excellent+model+category for the definition), why are all maps from the initial object of $\mathbb{S}$ monomorphisms?
Of course, I would also like to know what are the minimal conditions one nned to impose on a category with an initial object in order for every map originating from it to be a monomorphism.