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Is there a notion of Reedy cofibration category written down somewhere ? I don't want to reinvent the wheel. Motivation: I ask the question because I have a cofibration category in the sense of http …

Consider the category of elements construction described in https://ncatlab.org/nlab/show/category+of+elements. It induces a left adjoint from $[\mathcal{C},\mathrm{Set}]$ (the category of functors fr …

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 consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category …

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 …

Consider a fixed proper simplicial combinatorial model category $\mathcal{M}$. Consider a functor $F:I\to J$ between small categories. It induces a right Quillen functor $F^*:\mathcal{M}^J \to \mathca …

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 …

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)

Let $X:I\to M$ be a functor from a small category $I$ to a simplicial accessible (not combinatorial) proper model category $M$ which is already injective cofibrant. Let $W:I^{op}\to \mathrm{Set}$ be a …

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 …

Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]_0$. We work with the category of $\Delta$-generated spaces equipp …

It is a follow-up of my question Calculation of the homotopy colimit of a diagram of spaces which was badly formulated. Consider a fixed diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is a conveni …

Consider a small category $I$. There exists a small diagram $D:I^{op}\to {\rm Top}$ where ${\rm Top}$ is a convenient category for doing algebraic topology such that for all small diagrams $X:I\to {\r …

There is a well-known theorem for transporting a model category structure along a left adjoint $F:\mathcal{M}\to \mathcal{N}$ which is explained here and which is due to Sjoerd Crans. The difficult p …