Let $(\infty,1)Cat^{*/}_{con}$ denote the $(\infty,1)$-category of pointed and connected $(\infty,1)$-categories (that is, $(\infty,1)$-categories with a distinguished object such that there exists a zigzag of arrows between every two objects). Is there a model category presentation of this? As I recall, the model structure on reduced simplicial sets models the $(\infty,1)$-category of pointed and connected $\infty$-groupoids, which is slightly similar to my situation. Perhaps there is a related notion?

  • $\begingroup$ Model categories model (∞,1)-categories. How exactly is a model category supposed to model noninvertible 2-morphisms? $\endgroup$ – Dmitri Pavlov Aug 10 '16 at 20:11
  • $\begingroup$ Sorry, mistake on my part. We should restrict to the $(\infty,1)$-category of pointed and connected $(\infty,1)$-categories - I'll edit the question. $\endgroup$ – user84563 Aug 10 '16 at 22:37
  • $\begingroup$ How is a pointed connected (∞,1)-category defined in the first place? Can it have more than one object? The definitions I know of (see the nLab) say that a pointed category is simply a category with a zero object and a connected category is a category where any pair of objects can be connected by a zigzag of morphisms. Is this correct? $\endgroup$ – Dmitri Pavlov Aug 11 '16 at 7:24
  • $\begingroup$ @user84563 Note that a pointed space is not pointed when consider as an $(\infty,1)$-category (yeah, the terminology conflict is annoying but we're stuck with it). However in your notation you seem to suggest that by pointed category you mean a category with a distinguished object and not a category with a (non necessarily specified) zero object). Can you clarify the terminology? $\endgroup$ – Denis Nardin Aug 11 '16 at 8:46
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    $\begingroup$ I've clarified in the question. Is the clarification sufficient? $\endgroup$ – user84563 Aug 11 '16 at 12:54

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