There are many ways in which the categories of simplicial sets and of simplicial objects work very well, as mentioned above. More recently there has been a revival of the use of cubical sets, but with the additional structure of connections, derived from the monoid structures $\max,\min: I^2 \to I$. Dan Kan's first paper was cubical, but it was then realised that cubical groups were not Kan complexes, and there was a serious problem with realisation of cartesian products. In 1996 A. Tonks proved cubical groups with connections are Kan and G. Maltsiniotis has proved that cubical sets with connections form a test category in the sense of Grothendieck.

Thee is more on cubical sets in my answer to this mathoverflow question, in the book Nonabelian Algebraic Topology, and in this recent article.

There are lots of areas which have been well worked over in the simplicial context but not in the cubical (with connection) context, e.g. cubical groups. So there is lots more evaluation work to be done.

There are many ways in which the categories of simplicial sets and of simplicial objects work very well, as mentioned above. More recently there has been a revival of the use of cubical sets, but with the additional structure of connections, derived from the monoid structures $\max,\min: I^2 \to I$. Dan Kan's first paper was cubical, but it was then realised that cubical groups were not Kan complexes, and there was a serious problem with realisation of cartesian products. In 1996 A. Tonks proved cubical groups with connections are Kan and G. Maltsiniotis has proved that cubical sets with connections form a test category in the sense of Grothendieck.

Thee is more on cubical sets in my answer to this mathoverflow question, in the book Nonabelian Algebraic Topology, and in this recent article.

There are lots of areas which have been well worked over in the simplicial context but not in the cubical (with connection) context, e.g. cubical groups. So there is lots more evaluation work to be done.