Leaving rigorous definitions to the talk, "operator categories" are an umbrella concept for things built from Hilbert space operators, including C*-algebras, operator systems, hereditary manifolds, operator algebras, Jordan operator algebras, etc. I will show how to associate the following three features to any such category: a topology, a representation theory, and a convexity/dilation theory. It turns out that if one of these features agrees for a pair of categories, then all three do, in which case the categories are called equivalent. I will discuss some equivalences, along the way obtaining new observations about Arveson's hyperrigidity and maybe even triangles.