17 - Duality

Summary

We can regard all arrows in a category as pointing the other way, and this gives us the dual category. One advantage is that we immediately get a dual version of every construction and every theorem. We begin by exploring some small examples such as categories of factors, and turn all the arrows round to see what the resulting structure looks like. Thus motivated, we make the definition of dual category, and explain that any categorical structure has a dual version which is given by placing that structure in the dual category. We show that in this sense monics and epics are dual, and that isomorphisms are self-dual. We also describe the concept of duals of results, which are found by placing the result in the dual category. We show that the composite of two monics is monic, and that the dual result is that the composite of two epics is epic; we also consider the converse. We show that terminal and initial objects are dual. Finally, we briefly mention how the notion of dual category comes from symmetry in the definition of a category, more easily seen from the definition as an underlying graph with extra structure.