Two of my favorite categories show that they really are everywhere: the free category on any graph and the presheaves of gamma.
The first: take any directed graph, unfocus your eyes and instead of arrows consider paths. That is a category!
The second: take any finite graph. Take sets and functions that realize this graph. This is a category, moreover you can make it dagger-compact, so you can do quantum mechanics with it. Take as the finite graph gamma, which is just two vertex with two arrows between them. Sets and functions that realize this graph are… any graph! So, CT allows you to do quantum mechanics with graphs.
Two of my favorite categories show that they really are everywhere: the free category on any graph and the presheaves of gamma.
The first: take any directed graph, unfocus your eyes and instead of arrows consider paths. That is a category!
The second: take any finite graph. Take sets and functions that realize this graph. This is a category, moreover you can make it dagger-compact, so you can do quantum mechanics with it. Take as the finite graph gamma, which is just two vertex with two arrows between them. Sets and functions that realize this graph are… any graph! So, CT allows you to do quantum mechanics with graphs.
Amazing!