realization of maps between classifying spaces of categories The classifying space $B\mathcal{C}$ of a small category $\mathcal{C}$ is by definition the geometric realization of the nerve of $\mathcal{C}$. Now let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two small categories. Any functor $F : \mathcal{C}_1 \rightarrow \mathcal{C}_2 $ induces a continuous map $BF: B\mathcal{C}_1 \rightarrow B\mathcal{C}_2$.
My question is the following: when is a continuous map $f: B\mathcal{C}_1 \rightarrow B\mathcal{C}_2$ homotopic to a map of the form $BF$, for some functor $F:\mathcal{C}_1 \rightarrow \mathcal{C}_2 $
Thanks.
 A: Here's a counterexample. Let $M$ be a discrete monoid which is not a group. Consider the 
map
$$
M = \hom (\Bbb N, M) \to \text{maps}_{*}(B\Bbb N , BM) = \Omega BM
$$
($\Omega BM =$ the based loops of $BM$ which is the same thing as
Segal's group completion of $M$). 
This map is not a $\pi_0$ surjection since $M$ isn't a group. 
Now, this fits into the context of your question since $\Bbb N$ and $M$ may be
considered as categories with one object and a functor is just 
a homomorphism (i.e., $\cal C_1 = \Bbb N$ and ${\cal C}_2 = M$).
Remark: In the case when $\cal C_1$ and $\cal C_2$ are both groups, it would appear to me that 
the question you are asking has an affirmative answer.
A: Here's one instance where the answer is affirmative: when the target category is a groupoid (a little more general than my remark on John Klein's answer).
Let $\mathcal{C}$ be any category and $\mathcal{G}$ be a groupoid. Then $B\mathcal{G}$ is a 1-type (i.e., all $\pi_n$ with $n\ge 2$ vanish, with any base points), and therefore, for any space $X$, $\mathrm{maps}(X, B\mathcal{G})$ is a 1-type too, and in fact is just weakly homotopy equivalent to $B \mathrm{Fun}(\pi_{\le 1}(X), \mathcal{G})$ where $\pi_{\le1}(X)$ is the fundamental groupoid of $X$ (notice that since $\mathcal{G}$ is a groupoid, so is any functor category $\mathrm{Fun}(\mathcal{A}, \mathcal{G})$). Applying this to $X=B\mathcal{C}$, we get a weak homotopy equivalence $\mathrm{maps}(B \mathcal{C}, B \mathcal{G}) \simeq B \mathrm{Fun}(\pi_{\le 1}(B\mathcal{C}), B\mathcal{G})$. Now, $\pi_{\le 1}B \mathcal{C}$ is just $\mathcal{C}[\mathcal{C}^{-1}]$, the localization of $\mathcal{C}$ obtained by adding formal inverses for all morphisms of $\mathcal{C}$; and since $\mathcal{G}$ is a groupoid, any functor $\mathcal{C} \to \mathcal{G}$ automatically factors through $\mathcal{C}[\mathcal{C}^{-1}]$, so composition with the canonical functor $\mathcal{C} \to \mathcal{C}[\mathcal{C}^{-1}]$ induces an equivalence of categories $\mathrm{Fun}(\mathcal{C}, \mathcal{G}) \cong \mathrm{Fun}(\mathcal{C}[\mathcal{C}^{-1}], \mathcal{G})$. Equivalences of categories induce homotopy equivalences of classifying spaces, so we get a weak equivalence $\mathrm{maps}(B \mathcal{C}, B \mathcal{G}) \simeq B\mathrm{Fun}(\mathcal{C}, \mathcal{G})$, which is a little better than just saying that every function between the classifying spaces is homotopic to one coming from a functor.
