Scott, your snake equation proof looks good, except that you only show that the two profunctors are equal along the diagonal. That is, you only show that for all \(x\),
[ (1_X \times \cup_X ) (\cap_X \times 1_X )((1,x),(x,1)) = 1_X(x,x) ]
You need t…
But Puzzle 209 is about something more specific.
So, let me give some hints.
In Puzzle 209 the feasibility relation we're calling \(\Phi\) is just any old feasibility relation from \(\mathbb{N}\) to \(\mathbb{N}\): it's a very famous on…
John, so far information theory and statistics have been most applicable to my new work. I've been having fun following the course and have already had a few of the supernova concept collapse moments of the kind you mentioned in Lecture 17.
James, I studied physics in grad school. I was always on the more theoretical side, but I didn't actually become a mathematician until after grad school, when I was a postdoc fellow.
The dual lattice doesn't seem to be closed either. If we know "how to relate 1 to 2" and we want to know "how to relate 1, 2, and 3", we can either learn "how to relate 1 to 3" or "how to relate 2 to 3". But if you told me that all you know is eithe…
I was doing some reading to catch up with the class and I learned about the Adjoint Functor Theorem in Lecture 16. So now I know how to prove that \(R_3 = \mathrm{Ob}:\mathbf{Poset}\to\mathbf{Set}\) doesn't have a right adjoint.
We need to show tha…
Puzzle 194. From Lecture 11 we know that for any set \(X\) the set of partitions of \(X\), \(\mathcal{E}(X)\), becomes a poset with \(P \le Q\) meaning that \(P\) is finer than \(Q\). It's a monoidal poset with product given by the meet \(P \wedg…
Bat, yes, and because \(L_1\) and \(L_2\) drop objects when acting on non-discrete things, I was worried the composition wouldn't end up giving back Disc, which doesn't drop objects. But it turned out the resolution is that \(L_3\) only spits out di…
I had worked out a solution of the original version of Puzzle 163, even though it later turned out not to be intended as stated.
Puzzle 163. Show there are functors
[ \mathbf{Cat} \stackrel{R_1}{\longrightarrow} \mathbf{Preord} \stackrel{R_…