This question might have an easy answer, but my research is far from the region of topology that makes use of classifying spaces of categories, so I can't find it.

For (possibly infinite) integers $0 \leq k_1 \leq k_2 \leq \infty$, define a category $X(k_1,k_2)$ as follows. There are objects $S_n$ for any finite integer $n$ satisfying $k_1 \leq n \leq k_2$. The morphisms from $S_n$ to $S_m$ are maps $f:\{0,\ldots,n\} \rightarrow \{0,\ldots,m\}$ such that $f(i) < f(j)$ whenever $i < j$.

I remark that for $k_1=0$ and $k_2 = \infty$, this is almost the usual category of simplices. The only difference is that we require strict inequality $f(i) < f(j)$ in our morphisms rather than merely inequality $f(i) \leq f(j)$.

The classifying space of the category $X(k_1,k_2)$ is clearly $(k_2-k_1)$-dimensional. My question is whether or not this classifying space is $(k_2-k_1-1)$-connected? I can verify by hand that it is connected if $k_2-k_1 \geq 1$ and simply-connected if $k_2-k_1 \geq 2$, but my techniques are too ad-hoc to deal with the general case.

Thanks!