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Let $\mathcal{M}$ be a locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain …

sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual exam …

Some of the examples of $\infty$-categories are those arising from model categories. I would like to ask: what is the relationship between Quillen adjunctions between model categories and adjoint pair …

Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not. My question is: i) which of the 13 prob …

I am reading about the Reedy model structure from Hovey's book and I was wondering if the Reedy model structure on $\mathcal{M}^{\Delta}$ is cofibrantly generated by a small set of arrows and permits …

Let $\gamma : \mathcal{C} \to \mathcal{M}$ be a functor and define a cosimplicial resoultion of $\gamma$ as a functor $\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$ such that $\Gamma C$ is Reedy cofi …

It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it. I'm sure this can be done not only for $\text{Se …

Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$. Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\ …