[1]: http://www.ncatlab.org/nlab/show/simplicial+T-complex

This result has a long history. 

Keith Dakin in his 1976 thesis defined the notion of simplicial $T$-complex as a simplicial set with in each dimension $n \geqslant 1$ a set $T_n$ of elements called _thin_ such that 

1. Degenerate elements are thin. 

1. Every horn has a unique thin filler. 

1. If all but possibly one face of a thin element are thin, then so also is the remaining face. 

Such a $T$-complex is of rank $\leqslant n$ if all elements of dimension $>n$ are thin. 

Keith showed that simplicial $T$-complexes of rank 1 are equivalent to groupoids, and those of rank 2 are equivalent to crossed modules (over groupoids). The story was completed by Nick Ashley in 1978 who showed that simplicial $T$-complexes are equivalent to crossed complexes. Full references are given on the [nlab][1]. The corresponding result for groupoids  had an earlier airing in the paper:


Levi, F. W. Darstellung der Komposition in einer Gruppe als Relation. Arch. Math. (Basel) 8 (1957), 169–170. 

Update: I recommend you draw a diagram of a 3-simplex, label all the edges, interpret "thin" for a 2-simplex to mean "commutative boundary", and then consider the third axiom for thin elements in relation to associativity. This connection of the tetrahedron with associativity is well known in homological algebra.