What's an initial object in a poset-enriched category?

Question

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?

Answer 1

There are several possible definitions of initial object in a 2-category $\mathfrak{K}$; which one is appropriate depends on your applications.

1. A 2-category has an underlying ordinary category, so we may just reuse the standard definition of initial object.

2. A 2-category can be regarded as a category enriched over categories, so we may use the definition of initial object in an enriched category. Concretely, this refers to an object $a$ such that the hom-category $\mathfrak{K} (a, b)$ is the terminal category for all objects $b$ in $\mathfrak{K}$.

   Clearly, every enriched initial object is an initial object in the underlying ordinary category, but the converse is not true. (For example, the unique object of $\mathfrak{K} (a, b)$ may have non-trivial endomorphisms.) This definition is standard: see e.g. [Kelly, Basic concepts of enriched category theory].

3. We can do things up to isomorphism in a 2-category, so we might define an initial object to be an object $a$ such that the hom-category $\mathfrak{K} (a, b)$ has only one object up to isomorphism. This is the same thing as an initial object in the "homotopy category" $\operatorname{Ho} \mathfrak{K}$ obtained by identifying all isomorphic morphisms in $\mathfrak{K}$.

   Clearly, an object that is initial in the underlying ordinary category of $\mathfrak{K}$ is also initial in this sense, but the converse is not true. As far as I know, this definition is not used.

4. Every 2-category is also a bicategory, so we can take the definition from there. A bicategorical initial object in $\mathfrak{K}$ is an object $a$ such that the hom-categories $\mathfrak{K} (a, b)$ are contractible groupoids. More concretely, this means there is a morphism $a \to b$ for every $b$ and there is a unique 2-cell between any parallel pair of morphisms $a \to b$.

   Every bicategorical initial object is also initial in the sense of (3). This definition is a special case of the general notion of bicolimit: see e.g. [Kelly, Elementary observations on 2-categorical limits].

5. Every 2-category can be regarded as a simplicially enriched category by replacing each hom-category with its nerve. We could therefore define an initial object in $\mathfrak{K}$ to be an object $a$ such that the nerve of $\mathfrak{K} (a, b)$ is contractible for all $b$.

   It is not hard to see that bicategorical initial objects are also initial in this sense, but the converse is not true. (For example, $\mathfrak{K} (a, b)$ might have non-invertible morphisms.) I have not seen this definition before, but it may be useful if $\mathfrak{K} (a, b)$ is actually standing in for a homotopy type.

If the hom-categories of $\mathfrak{K}$ are posets (and I mean partially ordered set, not preordered set) then definitions (1) – (4) coincide. This is because the only isomorphisms in a poset are the identities.

Answer 2

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.