# FP on simplicial groups

It’s a good exercise to create fun problems (FPs). Here’s one:

Show that every simplicial group is a Kan complex.

In case you aren’t familiar with the terminology, let me explain. A simplicial group is a functor $\mathbf{\Delta}^{op}\to\mathbf{Grp}$ where $\mathbf{\Delta}$ is the simplex category. A Kan complex is a simplial set $X$ (i.e., a functor $\mathbf{\Delta}^{op}\to\mathbf{Set}$ such that for any $f:\Lambda^n_k\to X$, there is a lifting:

Here $\Delta^n$ is the $n$-simplex (try to define it as a simplicial set if you haven’t seen this before), and sitting inside it is the $k$th horn, which is obtained by deleting the interior and the face opposite the vertex $k$ in $\Delta^n$ (i.e., the cone centered on the $k$th vertex).