The diagrams TheCatsters are talking about seem to be one level above (functors between categories); while string diagrams Seven Sketches talks about refer to morphisms between objects.
Yes, I had the same thought. In video 4 or 5, cap and cup …
First, we generally write \(A \otimes B\) for the general monoidal product in \(f : A \otimes B \rightarrow C\) although the Cartesian product \(\times\) can be a monoidal product in \(\mathbf{Set}\). The monoidal product isn't the same as the cart…
To help me digest these ideas, I thought I'd write a summary of Tai-Danae Bradley's booklet. I've added my own reactions and I've tried link the ideas to what we've discussed in this course.
Two Themes
Tai-Danae starts with two basic themes in app…
To help me digest these ideas, I thought I'd write a summary of Tai-Danae Bradley's booklet. I've added my own reactions and I've tried link the ideas to what we've discussed in this course.
Two Themes
Tai-Danae starts with two basic themes in app…
I've never been in a Linear algebra, how does one calculate the dual of a vector space? And what is it used for?
Once you've chosen a basis, the dual of a vector is its transpose \(v^T\). This is because \(f(x)=v^T x\) is a linear functional on…
To be precise, there will be 1000! bijections between them. But there will be one "best" bijection.
Just for clarity, John got that monstruous number by computing 1000! (that's 1000 factorial). Why is the number of bijections 1000! ? For any fi…
I omitted subscripts for α and 1: I assume this is fine because they can never be ambiguous. Am I right in this assumption?
We write subscripts for \(\alpha\) since \(\alpha\) is really a natural transformation which transform one functor (doin…
While I am not an expert, I have worked with some people in hydrology. The traditional way to study hydrographs is to model responses as a linear time invariant system. Essentially, output signals like stream flow are related to input signals like p…
I think the course is going well! I wish I had more time to spend on the puzzles. Here's what I like
Starting with posets and monoidal posets, and adjoint monotone functions really helped me grasp the later ideas. Just today I was using the poset …
According to nLab:
If \(\mathcal{C}\) is a 2-category, there is its double category of squares \(\mathrm{Sq}(\mathcal{C})\) whose objects are those of \(\mathcal{C}\), both of whose types of morphisms are the morphisms in \(\mathcal{C}\), and wh…
Yeah, I knew you would be assuming readers already have the background. When reading math I don't know much about, it can take a while for things to click. If I read something and don't understand everything, I keep reading to get what I can and com…
Hi John, thanks for the links! I should read Brendan Fong's thesis. I wish I could find a good summary of Howard Odum's work (that doesn't skimp on the math). Combining traditional community and population ecology ideas (like the rate equations in …
This is something one has to get used to when reading category theory. I've gotten so used to it that I forgot to ever point it out!
It took me a while and several attempts to understand natural transformations and how they relate to categories…
One little nuance: we're assuming we have a coproduct for every pair of objects, and an initial object. But we need to pick a specific choice of these to make \(C\) into a symmetric monoidal category, because the definition of symmetric monoidal …
So I also should probably show that we actually have naturality for the natural isomorphisms (not just that these morphisms exist, but that they form natural transformations). But the proof is straight forward, here's the unitors. Naturality here me…
Theorem: A category \(C\) with all binary coproducts (there exists a coproduct for every pair of objects) and an initial object \(\mathbf{0}\) is a monoidal category where the coproduct is the tensor product and the initial object is the unit. If fo…
Simon wrote
, so it looks like you're calculating the diagram without the cup and the cap at the ends, right?
That's what I did... I forgot to remove the tensor identities. Also I got lazy about writing out natural numbers, hence the X, Y, Z. …
Puzzle 226 Here's some answers
The category of groups, group homomorphisms, with the direct sum (coproduct) as the monoidal product and the trivial group as the unit
Monoidal posets: a monoidal category where every homset is either a singleton or …
Here's a formula for this composite:
[f = (\mathrm{Id} \otimes \check \Sigma)(\mathrm{Id} \otimes \mathrm{Purchase}) (\mathrm{Id} \otimes \mathrm{Sell})(\mathrm{Id} \otimes \hat\Sigma)]
It has this type signature \(f: Z \otimes X \nrightarrow Z \o…
@John,
The subltety tripped me up a little at first since you'd written equality in the original post, but the type signatures don't work out to be the same, only isomorphic! Of course, sometimes we have equality and sometimes we only have isomorph…
One way to think about this \(f\) is to imagine it represents a machine or process that turns Z into X. So given \(z\) amount of Z, \(f\) answers whether or not we can make \(x\) amount of X. But the machine requires electricity (Y), but also happen…
I think that's right, looping an output into an input works like this: Let \(f : X \otimes Y \nrightarrow Z \otimes Y\). Tensoring and composing with cap, cup, and identity profunctors allows you to feed the \(Y\) output into the \(Y\) input, and ge…
Thanks Yoav, I think I forgot the definition of the identity profunctor and was thinking it imposed equality between inputs and outputs, but of course that violates the definition of a profunctor.
On a stylistic note, I think 0 is better than 1 …
First snake equation.
We can show that the diagrams for the snake equation are equal in terms of feasibility relations by expanding the right hand side (RHS) and showing it equals the identity on \(X\).
[ \left(1_X \times \cup_X \right)\left(\cap_…
Keith wrote
Actually, now that I think about, isn't
[ \cap_X(x,x') =\begin{cases}\mathrm{true} & \mathrm{if} \ x \le x' \\ \mathrm{false} & \mathrm{otherwise} \end{cases}]
a bit redundant? \([x \leq x']\) is already the relatio…
I learned something about 'muscle' from the physicist Hans Bethe
TIL that "Bethe" is spelled "Bethe", and not "Beta". I can't remember seeing the name written down before, mostly heard it in audiobooks about Feynman's life.