Post #3162807

Every week I have a meeting which turned into 'cute computations in presheaf topoi' and it's just the most satisfying time. Today we figured out what the adjoints to the 'Cayley graph' functor going from N-sets to Graphs are. The right adjoint is very easy (its infinite paths in the graph with the shift action), the left action is quite confusing but very satisfying to prove. It's fun to see the math solve its own problems! A trick we end up using a lot is: represent functors as change of base along a map of sites, write the coend formula as a left adjoint, then compute the coend as the quotient of a big coproduct. This way you can easily define maps out of this object, by defining them on the coproduct and check that they respect the newly added equations. The real art is to find a way to present the resulting quotient. In this case, can you figure out what it is?