Philippe Gaucher
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Registered User
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Mathematics and theoretical computer science. Interests: homotopy theory, category theory, locally presentable category, concurrency theory, process algebra, computer science.
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May 10 |
answered | Cofibrant replacements of a given object in a combinatorial model category |
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Apr 30 |
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Cofibrant replacements of a given object in a combinatorial model category It is yes if the category of cofibrations is accessible... I have no more information. |
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Apr 30 |
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Cofibrant replacements of a given object in a combinatorial model category I 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... |
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Apr 30 |
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Cofibrant replacements of a given object in a combinatorial model category I 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 ? |
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Apr 30 |
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Cofibrant replacements of a given object in a combinatorial model category Indeed, 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 ? |
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Apr 30 |
awarded | ● Commentator |
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Apr 30 |
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Cofibrant replacements of a given object in a combinatorial model category There 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". |
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Apr 30 |
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Cofibrant replacements of a given object in a combinatorial model category I 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. |
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Apr 29 |
awarded | ● Editor |
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Apr 29 |
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Cofibrant replacements of a given object in a combinatorial model category I added the reason in my question. |
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Apr 29 |
revised |
Cofibrant replacements of a given object in a combinatorial model category added 475 characters in body |
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Apr 29 |
asked | Cofibrant replacements of a given object in a combinatorial model category |
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Apr 29 |
accepted | Left determined model structure on delta-generated topological spaces |
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Apr 25 |
asked | Topological question about right-lifting property and the evaluation map |
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Mar 7 |
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About the Cole-Ström model category structure with a locally presentable category The 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). |
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Mar 7 |
asked | About the Cole-Ström model category structure with a locally presentable category |
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Dec 18 |
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The Quillen model structure on simplicial sets as a Bousfield localization Here 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. |
