Overcategories and undercategories of cofibrantly generated model categories (Q2063069)

Let \(\mathcal{M}\) be a model category and let \(Z\) be an object of \(\mathcal{M}\). The author proves that if \(\mathcal{M}\) is cofibrantly generated, cellular, left proper, or right proper, then the model categories \((\mathcal{M}\downarrow Z)\) and \((Z\downarrow\mathcal{M})\) are both cofibrantly generated, cellular, left proper, or right proper. This paper is a complement of the author's book [Model categories and their localizations. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.55001)].