TITLE: What's an initial object in a poset-enriched category?
QUESTION [7 upvotes]: I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the presence of initial objects in all the fibers $d\setminus F$ and use the 2-categorical version of Quillen's Theorem A due to Bullejos and Cegarra (see their pdf here). It'd be nice if these fibers had initial elements, so

What is an initial object $i$ in a 2-category $C$, and where can I find a reference in the literature?

I imagine that instead of having a unique morphism $1 \to c$ for every object $c$ of $C$ like one does for a 1-categorical initial object, we'd now want the category of all morphisms from $1$ to $c$ in $C$ to be contractible. So if $C$ is poset-enriched it would suffice for the poset $C(i,c)$ to have a minimal element for all objects $c$. Is this accurate, and if so, what can I cite as a reference?

REPLY [3 votes]: Dimitri Ara has just brought this question to my attention. Perhaps you will find the following useful. 
Let us say that an object $z$ of a $2$-category $\mathcal{A}$ has a terminal object if, for every object $x$ of $\mathcal{A}$, the category $Hom_{\mathcal{A}}(x,z)$ has a terminal object. (In particular, this category is non-empty.) 
The terminology comes from the fact that a category has a terminal object in this sense, viewed as an object of the $2$-category of categories, if and only if it has a terminal object in the usual sense. (This terminology was suggested to me by Bénabou because of that. See also Michael Barr's relevant answer to my question Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?) 
There are three dual definitions obtained by replacing $Hom_{\mathcal{A}}(x,z)$ by $Hom_{\mathcal{A}}(z,x)$ or "terminal" by "initial". The result below could of course be dualized accordingly. 
Denote by $e$ the trivial $2$-category. If a small $2$-category $\mathcal{A}$ has an object which has a terminal object, then the geometric realization of $\mathcal{A} \to e$ is a homotopy equivalence. This is Lemme 2.27 of http://www.math.jussieu.fr/~chiche/Maths/TheoremA.pdf. I still have to upload the final version on Arxiv. This is to be published in TAC.
Not sure whether it answers the question you were asking but hope this helps nevertheless.