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## Comments

I think I had some trouble explaining the point of the cap and the cup... though I'm so used to them that they seem obvious to me.

`I think I had some trouble explaining the point of the cap and the cup... though I'm so used to them that they seem obvious to me.`

@John. I am still trying to get my head around cups and caps. When I succeed (I will!), I will add an explanation of mine, since this poses a big struggle for me. I think the main difficulty of explaining cups and caps is that they 'weirdly' tie to together opposite preorders and units - and they have a weird visual representation that breaks with the visual pattern of "a wire is always connected to a box and vice versa."

`@John. I am still trying to get my head around cups and caps. When I succeed (I will!), I will add an explanation of mine, since this poses a big struggle for me. I think the main difficulty of explaining cups and caps is that they 'weirdly' tie to together opposite preorders and units - and they have a weird visual representation that breaks with the visual pattern of "a wire is always connected to a box and vice versa."`

Daniel - it might (or might not) help to think about cups and caps in the category of finite-dimensional real vector spaces, where the cup is the obvious linear map

$$ \cup_V \colon V^* \otimes V \to \mathbb{R} $$ and the cap is the linear map

$$ \cap_V \colon \mathbb{R} \to V \otimes V^* $$ that sends the number \(1\) to the identity operator, using the fact that the \(V \otimes V^*\) can be seen as the space of linear operators on \(V\).

Whether this helps or not depends on how comfortable one is with these ideas in linear algebra. But since I'm a mathematical physicist, I learned huge amounts about tensor products of vector spaces and dual vector spaces before ever meeting profunctors... so I find it easiest to think of the category of profunctors as a mutant version of the category of finite-dimensional real vector spaces, with profunctors looking very much like matrices. All the basic features are analogous: this is why I keep saying

profunctor theory is to category theory as linear algebra is to set theory.`Daniel - it might (or might not) help to think about cups and caps in the category of finite-dimensional real vector spaces, where the cup is the obvious linear map \[ \cup_V \colon V^* \otimes V \to \mathbb{R} \] and the cap is the linear map \[ \cap_V \colon \mathbb{R} \to V \otimes V^* \] that sends the number \\(1\\) to the identity operator, using the fact that the \\(V \otimes V^*\\) can be seen as the space of linear operators on \\(V\\). Whether this helps or not depends on how comfortable one is with these ideas in linear algebra. But since I'm a mathematical physicist, I learned huge amounts about tensor products of vector spaces and dual vector spaces before ever meeting profunctors... so I find it easiest to think of the category of profunctors as a mutant version of the category of finite-dimensional real vector spaces, with profunctors looking very much like matrices. All the basic features are analogous: this is why I keep saying **profunctor theory is to category theory as linear algebra is to set theory**.`

Thank you for the explanation and analogy! My intuition in this area is pretty weak - but at least I can use my then-knowledge of profunctors to understand these linear maps once I learn about vector spaces! At the moment I am in a phase were instead of approaching my understanding barrier directly, I revisit chapter 4 again and again and understand one little puzzle piece a time, asking small questions here and there. So any explanation helps! Once I understand, I will add my own :)

Edit: Great, I think the dam broke! :D The missing pieces where how to transform one co-design diagram into another that has the same meaning (by a isomorphism), whether cups and caps can be drawn as regulary boxes and tensoring (I like to think about it as "glueing objects together but remembering which, how many and in which order" so \( X \otimes X^\ast \) is isomorphic but not equal to \( 1 \otimes X \otimes X^\ast \)).

Regarding cups&caps: I found the visual difference rather confusing, It really helped me to think of them as a box. Another concept that helped me was to think of them as creators and annihilators. I think they are hard to explain because they do not fit as intuitively into co-design diagrams as the other concepts and as a learner I found them not well motivated: "In order to avoid drawing a backwards arrow, we introduce two boxes, a wire, new notation and the concept of a singleton preorder!" But I do not see how to improve on the timing of cap&cup introduction. I find the course beautifully structured and your exposure is great! What would have helped me in hindsight would (1) be the introduction of caps&cups and singleton preorders in terms of boxes first before going to the standard notation and (2) having an example of what why a cup together with a cap are equivalent to a backwards arrow in terms of feasibility relation. (Your explanation of the cup and cap individually was clear to me, it was the combination of them too that tripped me up a bit).

Feels great to finally understand a concept :D Thank you very much for this class!

`Thank you for the explanation and analogy! My intuition in this area is pretty weak - but at least I can use my then-knowledge of profunctors to understand these linear maps once I learn about vector spaces! At the moment I am in a phase were instead of approaching my understanding barrier directly, I revisit chapter 4 again and again and understand one little puzzle piece a time, asking small questions here and there. So any explanation helps! Once I understand, I will add my own :) ---- Edit: Great, I think the dam broke! :D The missing pieces where how to transform one co-design diagram into another that has the same meaning (by a isomorphism), whether cups and caps can be drawn as regulary boxes and tensoring (I like to think about it as "glueing objects together but remembering which, how many and in which order" so \\( X \otimes X^\ast \\) is isomorphic but not equal to \\( 1 \otimes X \otimes X^\ast \\)). Regarding cups&caps: I found the visual difference rather confusing, It really helped me to think of them as a box. Another concept that helped me was to think of them as creators and annihilators. I think they are hard to explain because they do not fit as intuitively into co-design diagrams as the other concepts and as a learner I found them not well motivated: "In order to avoid drawing a backwards arrow, we introduce two boxes, a wire, new notation and the concept of a singleton preorder!" But I do not see how to improve on the timing of cap&cup introduction. I find the course beautifully structured and your exposure is great! What would have helped me in hindsight would (1) be the introduction of caps&cups and singleton preorders in terms of boxes first before going to the standard notation and (2) having an example of what why a cup together with a cap are equivalent to a backwards arrow in terms of feasibility relation. (Your explanation of the cup and cap individually was clear to me, it was the combination of them too that tripped me up a bit). Feels great to finally understand a concept :D Thank you very much for this class!`