Suppose I have a simplicial space $X\_{\bullet}$ *without* degeneracies (sometimes called semi-simplicial space or incomplete simplicial space). There still is a geometric realization $\lVert X \rVert$ of $X\_{\bullet}$, which only uses the face maps. What properties does this realization have?


* Does it still preserve products, i.e. is $\lVert X \times Y \rVert$ still (weakly?) homotopy equivalent to $\lVert X \rVert \times \lVert Y \rVert$?
* Do levelwise (weak) equivalences still induce a (weak) homotopy equivalence of the geometric realizations?


I know of the paper "Categories and Cohomology Theories" by Segal, where he defines this "fat" geometric realization in the appendix. Unfortunately, he proves the above properties by a comparison with another construction that takes the degeneracies into account. Is this the only way to prove this? Do the properties fail in case there are no degeneracies?



simplicial spaces without degeneracies
In brief:


For your first question, no. Let $X\_\bullet$ be any semi-simplicial space and $Y\_\bullet$ have a point in degree zero and be empty in every other degree. Then $\vert X\_\bullet \times Y\_\bullet \vert = X\_0$, which will not usually be equivalent to $\vert X\_\bullet \vert$.


For your second question, yes. This is always true for semi-simplicial spaces, and is not always true for simplicial spaces (It is in the case where the thick and thin realisations are equivalent, of course). To see this you prove that the maps on $k$-skeleta $\vert X\_\bullet \vert^{(k)} \to \vert Y\_\bullet \vert^{(k)}$ are equivalences by induction on $k$, using the push-out description of the $k$-skeleton from the $(k-1)$-skeleton, and the fact that
$$X\_k \times \partial \Delta^k \to X\_k \times \Delta^k$$
is a cofibration so that it is a homotopy push-out. Then you use that $\vert X^\bullet \vert = \mathrm{colim} \vert X^\bullet \vert^{(k)}$ and that each $\vert X^\bullet \vert^{(k-1)} \to \vert X^\bullet \vert^{(k)}$ is a cofibration so this is a homotopy colimit.


