Let $\mathrm{sSet}_0$ be the category of simplicial sets with a single zero cell, also known as reduced simplicial sets. It is a well known fact (due to Quillen) that $\mathrm{sSet}_0$ supports a Quillen model structure where the cofibrations and the weak equivalences are the same as in the standard model structure on $\mathrm{sSet}$. It is well-known that the model structure on $\mathrm{sSet}$ is cofibrantly generated, with (acyclic) cofibrations generated by boundary inclusions of simplices (respectively inclusions of horns into simplices). Is the structure on $\mathrm{sSet}_0$ cofibrantly generated?
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$\begingroup$ I think yes — this is a bousfield localization of the usual model structure on simplicial sets. $\endgroup$– Tim CampionCommented Jun 5 at 20:37
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$\begingroup$ Or rather it’s a reflective subcategory of such a bousfield localization, obtained by imposing a partial fibrancy condition. $\endgroup$– Tim CampionCommented Jun 5 at 20:51
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$\begingroup$ It is a coreflective subcategory, right? Does that always preserve cofibrant generation? What is an explicit set of generating acyclic cofibrations? $\endgroup$– Gregory AroneCommented Jun 6 at 2:29
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$\begingroup$ It’s reflective, and not coreflective. It’s clearly closed under limits but doesn’t contain the initial object. It’s a model for pointed connected spaces, not for connected spaces. $\endgroup$– Tim CampionCommented Jun 8 at 18:14
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$\begingroup$ According to nLab page, the inclusion of sSet$_0$ into sSet is reflective, while the inclusion into sSet$_∗$ (pointed simplicial sets) is co-reflective. ncatlab.org/nlab/show/reduced+simplicial+set. I guess the inclusion into sSet$_*$ is reflective as well. It is still not clear to me if any of these facts automatically implies cofibrant generation. $\endgroup$– Gregory AroneCommented Jun 8 at 19:34
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1 Answer
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Yes, the model structure on reduced simplicial sets is cofibrantly generated.
An explicit proof of this statement is given by Goerss and Jardine in Simplicial Homotopy Theory, the proof of Proposition V.6.2, which constructs both factorization systems using the small object argument for certain sets of generating (acyclic) cofibrations.
answered Jun 8 at 15:40
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$\begingroup$ Thanks. I notice that the codomains of the generating acyclic cofibrations given by Goerss and Jardine are not finite or $\omega$-small. If I am not mistaken, this prevents sSet$_0$ from being cofibrantly generated as defined by Rezk-Schwede-Shipley, though it is cofibrantly generated according to most standard definitions. Is that correct? $\endgroup$ Commented Jun 8 at 18:52
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2$\begingroup$ @GregoryArone: The additional conditions added by Rezk–Schwede–Shipley are that generating cofibrations have cofibrant domains (true in our case) and (co)domains of generating (acyclic) cofibrations are small relative to the regular I-cofibrations. The latter condition is true because the category of reduced simplicial sets is locally presentable, hence every object is small relative to all morphisms, a much stronger condition. $\endgroup$ Commented Jun 8 at 18:57
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