Thanks John! I really liked this puzzle because it brought to life one of my favorite tenets for thinking about math and science: Things are exactly their relationships to other things.
Puzzle 148.
Let \(G\) be the graph with one node (labeled Person) and one edge (labeled BestFriend) from Person to itself, and let \(\mathcal C\) be the free category on \(G\).
Let \(F\) be the database where \(F(\textrm{Person}) = \{\textrm{Alice…
I've got a wondering about how stuff, structure, and properties fits into databases.
In the example of transforming databases from Lecture 41, the functor that ignores departmental data is full and faithful but not essentially surjective. So it "fo…
Jonathan, I like how you make explicit the connection to group theory. Is this connection a result of this particular example (maybe the fact that \(G\) has only one node and one edge) or a more general phenomenon?
What does our theory of databases say about this question? (As opposed to what a psychologist might say about it.)
My understanding is that our theory of databases makes no restriction for a person being their own best friend. In the definition…
Is there a reason the cateogry \(\mathbf{Set}\) is the "chosen" category when we define databases? Will we get interesting properties that we wouldn't get if we just defined a database to be functor from \(\mathbf{Free}(G)\) to any category? Or to a…
I got a different answer from Keith for Puzzle 111 (although they both share the nice property of being a power of 4 :) )
Puzzle 111.
To make a database we just need to define a set for every node in the graph and a set map for every edge. John de…
Puzzle 110
Since paths of length 0 in \( \mathbf{Free}(G)\) are the identity morphisms of that category and functors preserve identities, paths of length 0 from node \(x\) to itself must map to the identity function from the set \(F(x)\) to itself.
I've been thinking up some examples of \( \mathbf{Cost}\)-functors. I was inspired by John's use of the term short map and its connection to the word short cut.
Example 1
Let \(\mathcal X \) be the \(\mathbf{Cost}\)-category where the \(\mathrm{Ob…
As I was reading, I got confused about what \(\mathcal X'\) refers to? Should \(F: \mathcal X \to \mathcal X'\)? Instead of \(F: \mathcal X \to \mathcal Y\)?
Puzzle 91
Let's start with the first part
Show how to make the product of symmetric monoidal posets into a symmetric monoidal poset.
Let \((X, \leq_X, I_X, \otimes_X)\) and \((Y, \leq_Y, I_Y, \otimes_Y)\). We can define a new symmetric monoid…
After reading Matthew's proof in #8, I think that a \(\mathbf{Cost}^{\text{op}}\)-category is equivalent to a \( (\{ \tt{true}, \tt{false}\}, \implies, \tt{false}, \tt{or}) \)-category, that Daniel described in #10 of Lecture 29.
Both Daniel and Ma…
These three types of functors correspond to stuff, structure, and properties!! So we have a correspondence between
only adding capabilities is the same as forgetting purely properties
only changing the underlying set and forgetting purely structur…
John, I have a question about your hint in #13.
You said that
Conversely, if our monotone map between posets does preserve all joins (resp. meets) then it does have a left (resp. right) adjoint. But in this case it's often just as easy to guess…
@Keith, \(2[\textrm{egg}] \leq [\textrm{egg}]\) would mean that we can get 2 eggs from exactly 1 egg, which also isn't a valid reaction in the baking monoidal preorder described in the lecture. So I don't think Johnathan's proof works even for the o…
I'm a little confused about how the bowl renewable resource works in \(\mathbb N [S]\). I claim that if \(x, x' \in \mathbb N [S]\) and \(x \leq x'\) then the number of bowls in \(x\) and in \(x'\) are the same. This is because the relation \(\leq\)…
Hi Jonathan, I have a question about how you are thinking about the relationships in \(\mathbb N[S]\) and \(\mathbb N[T]\). You say:
For any particular \(T\)-complex \(y\), the set of \(T\)-complexes less than it are all those with no more \(T\)…
Dan noted in a Lecture 26 that John switched the unit conditions for lax and oplax in this Lecture. Here is his post:
John
I was having a hard time proving puzzle 83
and I think it's because the unit conditions are the other way around for …
Puzzle 82 Again let \(X\) be the monoidal preorder whose elements are words of finite length. \(s \leq_X s'\) iff the length of word \(s\) is smaller than the length of word \(s'\). \(\otimes_X\) is string concatenation and \(1_X\) is the empty stri…
To remember the condition for lax vs. oplax monoid monotone functions, I'm testing out this mnemonic:
Lax - doing things separately is cheaper than doing them together
Oplax - doing things together is cheaper than doing them separately
Lax functio…
Puzzle 80 Let \(X\) be the monoidal preorder described by Anindya in Comment 9 of Lecture 22 . The elements of \(X\) are finite length words, the relation \(\leq_X\) is defined by word length, and \(\otimes_X\) is word concatenation. The map that t…
John: Yes your condition hold for both the bag cost and coupon examples. Also I can see how it slides right into the proof I gave in Comment 6. I wrote,
\[f(x) \otimes_Y f(y) \leq_Y f(x') \otimes_Y f(y'). \] \(f\) exactly preserves the tensor st…
Let me see if I get what you're saying about the difference between a Petri net and a wiring diagram!
Let's stick with the example where our resources are
\(H\)
\(O\)
\(H_2O\)
The Petri net makes \(\mathbb N[S]\) into a commutative monoidal …