I will make a guess here and claim that the dagger functor is actually a pair of functors:
[ \dagger : \mathcal{X} \rightarrow \mathcal{X}^{op} \text{ and } \dagger : \mathcal{X}^{op} \rightarrow \mathcal{X} ]
Were it not so, the statement
[ \dagge…
Consider a slightly modified problem where 3 contains an additional free morphism.
In this case there are two distinct natural transformations \(\alpha, \beta : F \rightarrow H\) where \(\alpha_3 = c \circ a, \beta_3 = c \circ b \).
It first s…
I see, some care must be taken when constructing a graph suitable for construction of a Free category.
When parallel paths are present in the graph the are assumed to be not equal unless there is an equation specifically stating otherwise.
The onl…
Puzzle 131. ... Again, it may help to draw a graph.
I wanted to see where the overcount in comment 13 came from.
6:6 of length 0 (the identity transformations)
6:6 of length 1 [black] (the transformation arrows on the graph)
5:6 of length 2…
Puzzle 134. For any category \(\mathcal{C}\), what's another name for a natural transformation \(\alpha : F \Rightarrow G\) between functors \(F,G: \mathbf{1} \to \mathcal{C}\)? Yet again there's a simple answer using concepts you've learned here…
Addendum base on comment 22.
The commuting squares of which \(G \overset{\alpha}{\Rightarrow} G\) is typical do exist.
[
\begin{matrix}
\textbf{1} && \textbf{3} \\
1_1 & \overset{G}{\rightarrow } & 2_3 \\
id_1 \downarrow & \Dow…
Puzzle 132. For any category \(\mathcal{C}\), what's another name for a functor \(F: \mathbf{1} \to \mathcal{C}\)? There's a simple answer using concepts you've already learned in this course.
[ F \in Obs(\mathcal{C} ) ]
Puzzle 133. For any…
Puzzle 130. Let \(\mathbf{3}\) be the free category on this graph: How many functors are there from \(\mathbf{1}\) to \(\mathbf{3}\), and how many natural transformations are there between all these functors?
In comment 11 the candidates
for n…
We can either do the multiplication or we can examine the graph and find the least expensive path between the specified nodes.
[
\begin{matrix}
\begin{array}{c | c c c c}
M_X^2 & A & B & C & D \\
\hline
A & 0 & 6 & 3 &a…
If we take \( \mathcal{V} \) to be \( \textbf{Bool} := (\mathbb{B}, \le_{Bool}, \text{true}, \wedge) \)
and \( \mathcal{X}(m, n) := m \le_X n \), \( \mathcal{Y}(m, n) := m \le_Y n \), \( \Phi(m, n) := m \le_{\Phi} n \) , then
[ \mathcal{X}(x' , x)…
Here I show that it is true via a truth table.
[
\begin{array}{c c c | c c | c c | c }
b & c & d & b \wedge c & ( b \wedge c ) \le d & c \Rightarrow d & b \le ( c \Rightarrow d ) & ( b \wedge c ) \le d = b \le ( c \Righ…
2) Write down a profunctor \( \Lambda : \mathcal{X} \nrightarrow \mathcal{Y} \) and, reading \( \Lambda(x, y) = \text{true} \) as
“my uncle can explain \(x\) given \(y\)”, give an interpretation of the fact that the preimage of
\( \text{true…
Inspired by matrix rig comment.
E := employee
D := department
S := string
m := manager
a := secretary [assistant]
d := department name
f := first name
w := works in
[
A^0 = \left(
\begin{array}{c | c c c}
s \rightarrow t & E & D…
In comment there are some ideas.
I would like to know the names of these ideas.
We have a monoidal skeleton-category
labeled \( \mathbf{Matrix}_{skel} \)
and two categories
\( \mathbf{Matrix}_{\mathbb{N}} \)
and
\( \mathbf{Matrix}_{multiset}\) …
To make the adjacency matrix notion painfully obvious.
[ \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) ]
[ \left( \begin{array}{c | cc} s \rightarrow t & x & y \\ \hline x & \lbrace f \rbrace & \lbrace g \rbrac…
The latest version of the text had a major relabeling of all labeled items.
I believe this was due to the inclusion of Equations in the labeling scheme.
So, it is a good thing but it will require renumbering everything.
I updated the labels for Ex…