Timeline for When does a cofibrantly generated model category have this factorization property?
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11 events
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| Nov 21 at 3:00 | comment | added | Frank | Actually I am really confused about the existence of $J$, when we have already had a finitely presentable category $\mathcal{C}$ with cofibrations (between finitely presentable objects) generated by $I$, and a Brown category of finite $I$-cell complex (where the cofibrations are the relative $I$-cell complexes). | |
| Nov 21 at 2:36 | comment | added | Simon Henry | It can be any class that satisfies the assumption of the theorem. But it is meant to be the class of trivial cofibrations between cofibrant $\kappa$-presentable objects. Typically in the situation I was talking about above where you have a Brown category J is the class of cofibrations that are weak equivalences. | |
| Nov 21 at 1:59 | comment | added | Frank | Sorry for my stupidity. May I ask you what will the class $J$ in Theorem 2.4 be in this case? Thank you so much! | |
| Nov 19 at 16:24 | comment | added | Simon Henry | ... Note that condition (iii') of the theorem is obtained from a relative cylinder by taking B=B' and C to be the relative cylinder $B \coprod_A B \hookrightarrow C \overset{\sim}{\to} B$ | |
| Nov 19 at 15:51 | comment | added | Simon Henry | I'm not sure I understood the question correctly but: Theorem 2.4 in the linked paper implies that if you start with a finitely presentable category and $I$ generating cofibrations between finitely presentable objects. Then if the category of finite $I$-cell complex is a Brown category where the cofibration are the relative $I$-cell complexes (which is closely related to having a factorization as in the question) then this extend into a left semi-structure on the whole category which is cofibrantly generated by $I$ and by the set of "trivial cofibrations" of the Brown category. | |
| Nov 19 at 14:27 | comment | added | Frank | So when we try to construct a semi-model category with that property as you described, we should consider whether the factorization axioms of the Brown category satisfying finiteness (the first cofibration in the factorization is a relative (finite) $I$-cell complex) at first? | |
| Nov 19 at 14:26 | comment | added | Frank | Thank you so much for your answer! It really looks like what I want to know! I can see most of your points, except the $h_{\infty}$ part (I will read your paper seriously later). However, I have a stupid question. I am not sure that a cofibration between finite $I$-cell complexes is always a relative $I$-cell complex (if it is relative, the finiteness is automatic). | |
| Nov 19 at 14:14 | vote | accept | Frank | ||
| Nov 19 at 1:29 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 18 at 22:14 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Nov 18 at 22:09 | history | answered | Simon Henry | CC BY-SA 4.0 |