* The category of abelian group objects in the category SetsΔop of simplicial sets
* The category AbΔop of "simplicial abelian groups"
* The category of chain complexes
where Δ can be taken to be the category of all finite total orders, with (not strictly) monotone maps as morphisms. The major thing I didn't know was that chain complexes aren't just related to these other things, they're basically the same thing. Category theory, why you so crazy?