Dear Jonathan and David, Thanks for your answers and the references. This notion is what appeared naturally in our setting so it's nice to know that people have worked on it. We would very much like to hear more about what you (Jonathan) are working on in this respect. Sorry for the late letter but I had very bad internet access in the last week. Best, Adam On Sat, Jul 26, 2014 at 3:40 PM, Jonathan CHICHE 齊正航 <chichejonathan@gmail.com> wrote:

Dear Adam,

I don't have much time right now and have not read your paper carefully. However, to elaborate on David Roberts's answer, in my work about the homotopy theory of 2-categories, the property which I have found the most useful (and which shows up in many natural circonstances) is the following. Given a 2-category A, let us say that an object z of A has a terminal object if Hom(a,z) has a terminal object for every object a of A. This terminology was suggested to me by Jean Bénabou. It is of course compatible with the usual definition if the 2-category happens to be Cat. It can be shown that, if a small 2-category A admits such an object, then the map from A to the point is a weak equivalence, i.e. its nerve is a simplicial weak equivalence. I have some papers around related stuff, which I could communicate when they are in their final version.

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