Discrete simplicial spaces are fibrant

Question

As the title suggests, I would like to understand why should a discrete simplicial space be fibrant.

Let me be more precise. Consider the category $\textbf{sSet}^{\Delta^{op}}$ of simplicial spaces, endowed with the Reedy model structure. Define a simplicial space $W$ to be discrete if $W_n$ is a discrete simplicial set for each $[n]\in \Delta$, then the claim is that $W$ is fibrant with respect to such model structure.

Any help will be highly appreciated, thanks in advance.

Answer 1

One just has to calculate. Observe that:

1. Any morphism of discrete simplicial sets is a Kan fibration.
2. The full subcategory of discrete simplicial sets is closed under limits.
3. The relative matching objects are constructed using only (finite) limits.

It follows that any morphism of discrete simplicial "spaces" is a Reedy fibration. In particular, since the terminal object is a discrete simplicial "space", every discrete simplicial "space" is Reedy-fibrant.