# Philippe Gaucher

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## Registered User

 Name Philippe Gaucher Member for 11 months Seen 2 days ago Website Location Paris Age
Mathematics and theoretical computer science. Interests: homotopy theory, category theory, locally presentable category, concurrency theory, process algebra, computer science.
 May10 answered Cofibrant replacements of a given object in a combinatorial model category Apr30 comment Cofibrant replacements of a given object in a combinatorial model categoryIt is yes if the category of cofibrations is accessible... I have no more information. Apr30 comment Cofibrant replacements of a given object in a combinatorial model categoryI believed that "On combinatorial model categories" by J. Rosicky was the answer (math.muni.cz/~rosicky/papers/comb2.pdf), and page 8 (top of the page), one can read that $cof(S)$ is not always accessible, $S$ being a set ! On the contrary, $inj(S)$ is always accessible (Proposition 3.3 of the same paper) as a small injectivity class... Apr30 comment Cofibrant replacements of a given object in a combinatorial model categoryI cannot edit my comment so I rephrase my question. Take a "very good" cofibrant replacement $f.g$ of $X$ ($f$ cofibration and $g$ trivial fibration). Why does it come from a factorization by a functor $T$ constructed using the small object argument ? Apr30 comment Cofibrant replacements of a given object in a combinatorial model categoryIndeed, the class of accessible categories is closed under lax limits, not under limits. I do understand that your homotopy pullback proves that the class of composables maps (f,g) with $f.g=\varnothing \to X$ is accessible. But the intersection with the image of the functor $T$ does not answer the question. How do you choose $T$ ? Where does it come from ? Apr30 awarded ● Commentator Apr30 comment Cofibrant replacements of a given object in a combinatorial model categoryThere is a similar (but not sure that it is simpler) problem. In the category of $\Delta$-generated spaces, is the class of cofibrant contractible spaces accessible ? My intuition tells me "yes". Apr30 comment Cofibrant replacements of a given object in a combinatorial model categoryI don't understand your proof. Do you mean the map $\mathcal{C}^{[2]}\to \mathcal{C}^{[1]}$ which takes the composite ? And even with that, I still do not understand. By cofibrant replacement of $X$, I mean a pair $(X,f:Y\to X)$ where $Y$ is cofibrant and $f$ is a weak equivalence (I am also interested in the more restricted definition $f$ trivial fibration). And why use homotopy limits ? I believe (but I may be wrong) that the class of accessible categories, unlike locally presentable ones, is closed under a lot of operations like limits. Apr29 awarded ● Editor Apr29 comment Cofibrant replacements of a given object in a combinatorial model categoryI added the reason in my question. Apr29 revised Cofibrant replacements of a given object in a combinatorial model categoryadded 475 characters in body Apr29 asked Cofibrant replacements of a given object in a combinatorial model category Apr29 accepted Left determined model structure on delta-generated topological spaces Apr25 asked Topological question about right-lifting property and the evaluation map Mar7 comment About the Cole-Ström model category structure with a locally presentable categoryThe paper you mention even gives the answer: top of page 24 : "We should remark that any locally presentable topologically bicomplete category also satisfies our hypothesis". And as far as I can understand the paper, any convenient category of topological spaces is fine, by convenient it is meant cartesian closed and containing enough topological spaces (e.g. CW-complexes). Mar7 asked About the Cole-Ström model category structure with a locally presentable category Dec18 comment The Quillen model structure on simplicial sets as a Bousfield localizationHere is the beginning of an idea: by right-Bousfield localizing by all trivial fibrations, you will reduce the class of cofibrations. So the trivial fibrations must be interpreted as colocal equivalences. So I suggest first a right Bousfield localization by the set of simplices, and then if the new model category has exactly the monomorphisms as cofibrations (?), it should be "between" the minimal model structure and the usual model structure by Cisinski's result (so it should be left proper), then a left Bousfield localization by the accessible class of weak equivalences could work.