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 2morphisms? $\endgroup$ – 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

1$\begingroup$ I've clarified in the question. Is the clarification sufficient? $\endgroup$ – user84563 Aug 11 '16 at 12:54