Alexander Campbell's recent sequence of talks has 'put-to-bed' chain complexes by describing a category equivalent to strong Steiner complexes, without reference to chain complexes, and using it to establish precisely the relationship between the Steiner approach and Ross' parity complexes. I find this especially satisfying because I've wanted to see that relationship clearly, and because, after all, what have chain complexes, a computational tool for algebraic topology, really got to do with fundamental notions of category theory?
Actually, it turns out that the answer to that question is much more than at least some of us thought. This talk will outline the relationships (and include some history and some historical speculations).