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 where is the simplex category. A Kan complex is a simplial set (i.e., a functor such that for any , there is a lifting:
Here is the -simplex (try to define it as a simplicial set if you haven’t seen this before), and sitting inside it is the th horn, which is obtained by deleting the interior and the face opposite the vertex in (i.e., the cone centered on the th vertex).