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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 …

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 …

I read in nLab : Every cofibrantly generated model category structure can be lifted to that of an algebraic model category. It is not clear whether or not this is true for any accessible model categor …