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EDIT: First edit after an interesting answer. $(S,\mathcal{T}_1)$ and $(S,\mathcal{T}_2)$ are homotopy equivalent to the same Quillen cofibrant space. Let $S$ be a set with two topologies $\mathcal{ …

Take a Hausdorff topological space $X$. Take two distinct points $x$ and $y$ of $X$. Consider a set $U$ of continuous paths $p$ from $[0,1]$ to $X$ equipped with the compact-open topology such that: $ …

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 …

(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 …

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). …

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 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 …

It is the motivation of the question Examples of non Quillen-equivalent model categories having equivalent homotopy categories. I did not give at first the motivation because i don't think that people …

Consider the set of continuous maps $C^0([0,1],[0,1])$ equipped with the compact-open topology. It is metrisable, and therefore sequential. It is also a k-space: see http://neil-strickland.staff.shef. …

Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ …

Without AC, it is impossible to prove that every set is equipotent to an ordinal (in ZF !) and it is impossible to prove that a functor is an equivalence of categories if and only if it is full faithf …

It is not an answer and I cannot let something wrong: the isomorphism is a general fact about locally finitely presentable categories. Let $\mathcal{K}$ be a locally presentable category. The pullback …

I use the notation of this question. A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and …

Consider the discrete monoid $M$ of nondecreasing continuous maps from $[0,1]$ to itself preserving the extremities. Note that the monoid is right-cancellative ($x.z=y.z$ implies $x=y$, since $z$ is a …

1) QUESTION (EDIT: 04/28/2020 to remove a possible counterexample) I work with weak Hausdorff $k$-spaces (so all spaces are $T_1$). The internal hom is denoted by $\mathbf{TOP}(-,-)$. Let $\mathcal{G …