They are a machine for doing computations with a couple of fundamental steps. So you want to compute F(X), you first break down/approximate X by things where you can compute F (or something like it). You then turn a crank which is incredibly non0trivial (compute differentials etc). Finaly, you reassemble the pieces into what you want (also highly nontrivial). Perhaps this is all obvious though.

I thought Boardman and Vogt talked about weak Kan complexes, no? That was written before 1991. Although perhaps people don't appreciate that book. Dylan, want my netflix password? After this I would buy you a ticket to a film myself if I thought theaters were open.