We review a theorem of A. Roig about Quillen model structures on Grothendieck bifibrations and observe that it contains a gap. We reformulate one of its assumptions in order to validate it. As an application to the new version, we introduce the fibred model structure on the category of small categories enriched in a suitable monoidal model category.

We examine the proof of a classical localization theorem of Bousfield and Friedlander and we remove the assumption that the underlying model category be right proper. The key to the argument is a lemma about factoring in morphisms in the arrow category of a model category.

We establish a Quillen model category structure on the category of symmetric simplicial multicategories. This model structure extends the model structure on simplicial categories due to J. Bergner.

a b s t r a c tWe prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.

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