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In "Many homotopy categories are homotopy categories" (M. Cole / Topology and its Applications 153 (2006) 1084–1099), Cole generalizes the construction of the model category of Ström to any bicomplete …

Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying Pr …

Let $L:\mathcal{M}\leftrightarrows\mathcal{N}:R$ be a Quillen equivalence between combinatorial model categories such that all objects are fibrant. Let $X$ be a cofibrant object of $\mathcal{M}$. Then …

(sorry I have troubles with comments, I post here even if it is not an answer) I have a new information. In On a fat small object argument, it is proved that in a λ-combinatorial model category, every …

In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-filtered colimit of a d …

I would like to know how to cite this theorem (which has a quite surprising consequence): A model category $\mathcal{M}$ is right proper if and only if for any weak equivalence $f:A\to B$, the Q …

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback i …

By abstract construction of a combinatorial model category, I mean starting from a locally presentable category satisfying some assumptions, e.g. equipped with a cylinder or a cocylinder satisfying so …

I work with the category of $\Delta$-generated spaces. I call reparametrization category a small strict semimonoidal topologically enriched category $(\mathcal{P},\otimes)$ such that $\mathcal{P}(\ell …

Let $\mathcal{M}$ be a locally presentable category equipped with two left proper and left determined combinatorial model structures $\mathcal{M}_1$ and $\mathcal{M}_2$. There exist two sets $S_1$ and …

Consider a right Quillen adjoint which is not a categorical left adjoint which takes (trivial resp.) cofibrations between cofibrant objects to (trivial resp.) cofibrations between cofibrant objects. …

I need to prove the existence of a model structure but I am still unable to formulate a definition of the class of weak equivalences. I have the following informations: The underlying category is loc …

According to https://ncatlab.org/nlab/show/cellular+model, I call a cellular model of a locally presentable category a set of monomorphisms cofibrantly generating the monomorphisms. I am curious to se …

Let $\mathcal{M}$ be a model category and let $C:\mathcal{M}\to\mathcal{M}$ be a very good cylinder object. The natural transformations coming with $C$ are denoted by $\gamma^\epsilon_X:X\to CX$ with …

The isofibrations are the fibrations of the canonical model structure of the category of small categories. If I call fibration of small categories the same notion by removing the word isomorphism, i.e …