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:

horn.PNG

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 kth 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 kth vertex).

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