Q:

Example for geometric realization on semi-simplicial set that doesn't preserve limit

I'm looking for a diagram $D$ (as simple as possible) in the category of semi-simplicial sets (i.e $sSet$ with only monos) such that $R(\text{lim}\,D) \ncong \text{lim}\,R(D)$, where $R$ is the geometric realization on semi-simplicial sets. Moreover, in $sSet$ the equivalence should be true, i.e. I'm looking for an example for the necessarity of degeneracies.

A:

Take the empty diagram.  The terminal semisimplicial set has one simplex in each dimension, so its geometric realization is infinite-dimensional!
More generally, products of semisimplicial sets are rather poorly behaved.  For instance, if $X$ and $Y$ are $n$-dimensional semisimplicial sets  (meaning they have no simplices above dimension $n$), then so is $X\times Y$, so $R(X\times Y)$ is $n$-dimensional while $R(X)\times R(Y)$ is $2n$-dimensional.  For a very concrete example, consider when $X=Y$ is a $1$-simplex.  Then it is a good exercise to work out that $R(X\times Y)$ is a disjoint union of a $1$-simplex and two points.

