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Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the struct …

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent …

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a …

It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a recent result of Lennart Meier, a certain "double …

There is no difference for $\kappa = \aleph_0$. The point is that you can build colimits for filtered diagrams using just colimits for chains. Every filtered category $\mathcal{J}$ admits a cofinal …

Hirschhorn's book, Model categories and their localizations, is a very thorough reference with many basic results explicitly stated and proved. The result you want is implied by axiom SM7 for simplici …

I work in a general pointed model category. The homotopy fibre of a morphism $f : Y \to X$ can be defined as follows: first, choose a fibrant replacement $w_X : X \to \hat{X}$ and then factor $w_X \ci …

One of Quillen's original examples was the category of chain complexes of finitely-generated modules over a ring – this is obviously equivalent to a small category, and of course, one has to use Quill …

Yes, $D_0 \times_{D_1} D_2 \to (E_2 \times_{E_1} E_0) \times_{E_0} D_0$ is a fibration. First, observe that $$(E_2 \times_{E_1} E_0) \times_{E_0} D_0 \cong E_2 \times_{E_1} D_0 \cong (E_2 \times_{E_1} …

There are only nine possible model structures on $\mathbf{Set}$. You can easily check that none of them give the right weak equivalences: Either everything is a weak equivalence, or $X \to Y$ is a w …

Since you have functorial factorisations you should exploit that to the hilt. If $\mathcal{M}$ is a model category with functorial factorisations then the category $\mathbf{c}\mathcal{M}$ of cosimplic …

Here is a somewhat degenerate example that illustrates what can go wrong. Let $\textbf{Ab}$ be the category of abelian groups, considered as a homotopical category where the weak equivalences are the …

Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield …

Let $\mathcal{M}$ be a category, let $\mathcal{M}^\circ$ be a reflective (full) subcategory of $\mathcal{M}$, and let $L : \mathcal{M} \to \mathcal{M}^\circ$ be the reflector. Question. Does there exi …

There are many equivalent ways of defining the local equivalences. Let $\mathcal{M}$ be a simplicial model category with a cofibrant replacement functor $Q : \mathcal{M} \to \mathcal{M}$. Then, for a …