Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has anyone tried to find out at all, whether this model category is right proper? And, if yes, then produced a reference?


P.S.: I know that I can get a Quillen equivalent right proper model catgory by the techniques of Thomas Nikolaus, but in that setting it is hard to tell what a general cofibration looks like, which lets me shy away from that step...

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    $\begingroup$ I have nothing useful to contribute, I just wanted to suggest that any use of the phrase "right proper" should be accompanied by "guv'nor". $\endgroup$ – Nate Eldredge Oct 4 '10 at 2:39

The model structure for complete Segal spaces is not right proper. To see this, one can first prove that the model structure for quasi-categories is not right proper: for instance, the map $\delta^1_2:\Delta_1\to\Delta_2$ is a fibration between fibrant objects in the Joyal model category (because it is the nerve of a fibration of the canonical model structure on the category of small categories), but its pullback along the inner horn $\Lambda^1_2\to\Delta_2$ is the boundary $\partial\Delta_1$. Now, given a quasi-category $X$, there is a canonical complete Segal space $N(X)$ associated to it (denoted like this because this is an homotopic version of the classical nerve): the space of $n$-simplices of $N(X)$ is the Kan complex $Map(\Delta_n,X)$, where $Map$ means the mapping space for the Joyal model category structure. A nice explicit model for $Map(\Delta_n,X)$ is just $k(\underline{Hom}(\Delta_n,X))$, where $\underline{Hom}$ is the internal Hom in simplicial sets, and where $k(A)$ denotes the maximal Kan complex contained in the quasi-category $A$. For this explicit model of $N(X)$, if $X=\Delta_m$, then we get that $N(X)$ is just the classical nerve of the poset corresponding to $\Delta_m$ (because there are no other isomorphisms than the identity in $\Delta_m$). In other words, $\Delta_m$ is a complete Segal space already. Therefore, the counter-example for right properness given above for quasi-categories gives a counter-example for complete Segal spaces. For the same reason, the model structure for Segal categories is not right proper.

Edit: Just to avoid any further hope: I considered implicitely that we worked with the model structures for which the cofibrations are the monomorphisms. However, the notion of (right) properness only depends on the class of weak equivalences, so that there just no hope to get right properness for these homotopy theories of $(\infty,1)$-categories, even if we change the notion of fibration.

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    $\begingroup$ Thank you!! - especially also for destroying all further hope :-) I wasn't aware of the independence of properness of the model structure, but it's true and great to know. Prop. 2.7 in here provides a reference, for those who wonder: arxiv.org/abs/math/0003065 $\endgroup$ – Peter Arndt Oct 4 '10 at 17:29

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