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Let $\mathcal{K}$ be a locally presentable category. I recall that a cylinder $C:\mathcal{K}\to \mathcal{K}$ is by definition equipped with two natural maps $\gamma_X:X\sqcup X\to CX$ and $\sigma_X:CX …

Let $\mathcal{A}\subset \mathcal{B}$ be two categories with $\mathcal{A}$ full and reflective in $\mathcal{B}$. Let $R:\mathcal{B}\to\mathcal{A}$ be the reflection. That $R$ is the left adjoint to the …

With Yoneda ? For every object $X$, $Mor(a\oplus (-a),X)$ is a singleton since $a \oplus (-a)$ is initial. And $Mor(a\oplus (-a),X) \cong Mor(a,X) \times Mor((-a),X)$ is a singleton as well. So $Mor(a …

Let $\mathcal{K}$ be a category. I denote by $\mathcal{D}\mathcal{K}$ the category of all small diagrams over $\mathcal{K}$: an object is a functor $F:I\to \mathcal{K}$ from a small category $I$ to $\ …

It is another question about the category $\mathcal{D}\mathcal{K}$ of all small diagrams over all small categories, its definition is here : About the category of all small diagrams. I suppose $\mathc …

Consider a functor $F:\mathcal{A}\to \mathcal{B}$ between two small categories. Let $\mathcal{K}$ be a locally presentable category. Consider a functor $G:\mathcal{A}\to \mathcal{K}$, there is a natur …

The binary product of two $\lambda$-presentable objects (in a locally presentable category) is $\mu$-presentable for some regular cardinal $\mu \geq \lambda$ (because all objects are $\mu$-presentable …

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 …

Nobody gave this reference so I give it : http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html, "The joy of cats", especially chapter 21 p350. The notions of initial and final topologies are ge …

There are some graphic calculations in dimension 3 and 4 in my paper Combinatorics of branchings in higher dimensional automata (p348-358 in dimension 3 and p361-362 with a 4-cube).

I can answer my question now... Not only the Quillen model structure on $\Delta$-generated spaces is left determined, but also the hypothesis $\Delta$-generated can be removed. The left determined mod …

Concerning your third question, the cogenerators of the category of general topological spaces are precisely the non-$T_0$-spaces. See Example 7.18 Remark (4) in Adamek, Herrlich and Strecker's Abstra …

Consider the class of cofibrations of the Quillen model structure, restricted to delta-generated topological spaces (the full subcategory of topological spaces generated by the colimits of simplices). …

I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories. The motivation is of course that I have two model categories and all t …