I am trying to see if we can interpret an \(n\) by \(m\) real matrix \(\mathbf{A}\) as a functor so that its conjugate transpose \(\mathbf{A}^*\) is the right adjoint in the categorical sense and use it to understand the meaning of naturality. Howe…
Alright, I'm thinking out loud here, trying to cook up the functor \(\mathbf{Cat}\overset{R_1}\longrightarrow\mathbf{Preord}\) in Puzzle 163. Currently I am only thinking about how \(R_1\) acts on the objects. So let's pick a random category \(\ma…
@Keith Like I said, I suspect that John left out some technicality so that we can focus on the essential features of category theory. To be fully precise, we should define \(\mathbf{Cat}\) as the category of all small categories, which means the me…
@Keith comment38, the collection of objects \(\mathrm{Ob}(\mathbf{Set})\) in the category \(\mathbf{Set}\) is not a set, it is a class. This means \(\mathbf{Set}\) is a large category, echoing Anindya's comment#41. This is a way to resolve Cantor'…
Puzzle 162
Piggyback on Dan's answer in comment#20, I think one can make the answer shorter by combining the two cases there into one.
To avoid level slip, let's clarify the objects and morphisms on different levels before we start. To prove that…
Elaborate a bit more on Keith's answer.
Puzzle 161
1) Preservation of composition:
Suppose \(h\in\mathcal{C}(c, c')\) and \((f,g)\) is a morphism from \((c, c')\) to \((d, d')\) and \((l, j)\) is a morphism from \((d, d')\) to \((e, e')\), we hav…
Thanks Matthew. Would not be able to do it without the formula proposed by you and Jonathan. Collaboration at its finest here!
I am still working at Puzzle137. I thought the functors would form a totally ordered set with the lexicographical orde…
Puzzle 136
I understand Jonathan and Matthew had already answered this puzzle in comment 2 and comment 32, but I would like to fill in some details.
As Jonathan pointed out, functors from the category \(\mathbf{m}\) to the category \(\mathbf{n}\) …
Dear John, you wrote
2) Then we need to count the number of ways to chop our n-element set into k subsets of size p and n−kp subsets of size 1. The answer is the multinomial coefficient (np,p,…,p,1,1,…,1) with k p's and n−kp 1's.
Since the arr…
John wrote:
Puzzle. If p is prime, how many actions of the group Z/p are there on a set with n elements?
For a database with schema \(\mathbb{Z}_p\) and maps the node to a set with n elements, the edge is mapped to either the identity or an n-…
I am not a programmer so I may be saying something silly...
Puzzle 109 In practical terms, does it mean if you want to write a function to retrieve the name of the department that an instance of the Employee data works in as a string type, do not …
Puzzle 85
Suppose \(y\in\mathbb{N}(T)\), hence \(y=a_{E}[E]+a_{Y}[Y]+a_{W}[W]\) where \(a_E, a_Y, a_W\in \mathbb{N}\). I am not sure if I am correct but my observation is that in \(\mathbb{N}(T)\), even the reflexive property implies that
$$a_{E}…
The monoidal unit, call it \(I\), should be \(false\) because
\(false\vee I = I\vee false= false \vee false = false\)
\(true\vee I = true \vee false =true = false\vee true = I\vee true\)
Moreover,
\((false\vee false)\vee false=false=false\vee(…