@PeterGates Oh, It is a MathJax thing. At least part of the rendering happens locally. It takes some time to download the relevant javascript libraries. So, you will see a basic text representation followed by the properly rendered version. Once you…
This discussion is limited by the lack of names on the comments.
I thing a new "Category" maybe call it
'Applied Category Theory Samples'
to go along with 'Applied Category Theory Course' and
'Applied Category Theory Exercises'.
There have been a…
The economic models are "fun".
Consider the couch surfer, staying in one place too long will incur a cost but travelling is less costly.
The costs increase the longer you stay.
They expect you to pay for food and stuff.
Not quite sure how to deal w…
red-herring comment
Funny.
The poset vs. preordered set vs. partially-order set, discussion that happened earlier seems to be a case of the mathematical red herring principle almost happening.
When we start with a set, let's call it \( \mathbb{X} \…
Back to the metric space comment .
A metric space consists of
1) a set \(X\) the elements of which are called points and
2) a metric function \( d : X \times X \rightarrow \mathbb{R}_{\ge 0} \), where \( d(x,y) \) is the metric, a distance betwee…
Ok, I will make an edit, I figured that if it was called a category that must be what it was. :-P
Do we know that \( \mathcal{X} \) will be a preorder?
Doesn't that depend on the monoidal preorder \( \mathcal{V} \) that we select.
Are you saying…
Bool \( := (\mathbb{B}, \le, \tt{false}, \wedge) \).
The following is not the complete proof, it only shows the mapping of data and
not that the rules of the two concepts are honored.
Given a preorder \( \mathcal{P} : (P, \le) \) use it to define…
I was not understanding what was being 'enriched' and what was doing the 'enriching'. Based on comments I think I am not the only one. Here is my informal definition of \( \mathcal{X} \) a \( \mathcal{V}\)-enriched category or \( \mathcal{V}\)-cate…
I think the discussion is limited by the idiosyncrasies of formatting posts.
The availability of good examples for formatting comments and discussion headings.
Putting them in the Wiki does not really work because the rendering rules differ slightly…
You need to select the gear, in the upper right corner, and then 'view source'. You can copy that. Although you cannot delete a comment you can 'edit' it, again via the 'gear'.
Puzzle 71
I think the problem comes down to whether there is a mapping from
$$ \mathbb {C} \rightarrow \mathbb {R} \times \mathbb {R} \rightarrow \mathbb {R} $$
Clearly \( \mathbb {R} \) forms a nice poset.
One nice poset is the total order.
I kno…
Puzzle 66
@DanOneata
Where did the definition of the relation \(\le_{X \times X}\) in the product preorder come from?
$$ x \le_X x' \text{ and } y \le_X y' = (x, y) \le_{X \times X} (x', y') $$
Is it just an alternate formulation of the "pixie d…
Puzzle 62
An easy way to repair the failed monoidal preorder would be to throw away part of the real numbers.
$$ \textbf{R}^+ := ( \mathbb{R}^+, \le, 1, \cdot ) $$
discarding the negatives and 0.
Also, I think the difficulty pointed out by @Jonatha…
@JaredSummers My intuition told me to pick right hand values that are relatively prime.
Does that have any relevance or did I just get lucky.
In your example 4 and 6 share a common factor, 2, and so that gets rejected.
The pairs \( 2|4 , 5|15 \ri…
@ValterSorana You do not need negative numbers; I wanted to point out a concept related to an alternative opposite.
The alternative invokes the 'cost' set as a negative value.
I believe your interpretation of it as a simple change in perspective be…
We have for \( \mathcal{X} \) that
(b) \( \text{ for all } x \in X, \text{ the equations } I \otimes x = x \text{ and } x \otimes I = x \text{ hold } \),
(c) \( \text{ for all } x, y, z \in X, \text{ the equation } (x \otimes y) \otimes z = x \oti…
Proposition 2.20 implies that
$$ \textbf{Cost}^{op} = ( [ 0, \infty ], \le, 0, + ) $$
is a perfectly good symmetric monoidal preorder.
Using \( 0 \) for the monoidal unit, and \( + \) as the monoidal product seem fine.
Thinking of cost as a nega…
I have been making updates to the chapter 1 exercises for which I have edit rights.
I was not sure what you wanted to do with the pair of E66C1. Should the "preorder" "poset" note be added to the title?
Yes, in an effort to make the exercises self …
The approach suggested by Eq. 2.9 is that as a "cut line" moves from left to right as each inequality is "cut"
a inequality is expressed.
Put another way, the application of 2.1.a produces an implication.
Throughout the proof \( + = \otimes \).
Giv…
@ValterSorana Much of what I did in the proof was to get to E4: \(x \le g_0(y) \iff x \le g_1(y) \) [which is where you start]. From there out it is just detailing what you said. D0, D1, L0a, L0b, L1a, & L1b are just the theorems for adjunctions…