Timeline for Does the morphism of composition have some universal property?
Current License: CC BY-SA 4.0
15 events
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| Jun 19, 2021 at 10:41 | history | edited | Johan Thiborg-Ericson | CC BY-SA 4.0 |
Previous work reference.
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| Jun 18, 2021 at 6:05 | comment | added | Johan Thiborg-Ericson | Thanks @Buzz :) | |
| S Jun 18, 2021 at 2:39 | history | suggested | Buzz | CC BY-SA 4.0 |
converted to LaTeX
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| Jun 17, 2021 at 21:12 | comment | added | Johan Thiborg-Ericson | @Paul Taylor: I am totally fine with there being many arrows satisfying my condition. In category theory, one can never hope to get more specific than "up to isomorphism", and this goes for arrows with universal properties as well. I am pretty certain there are many arrows in exponential objects in Set that satisfy the same universal property as regular function evaluation. One that consistently permutates input or output for example. | |
| Jun 17, 2021 at 20:46 | review | Suggested edits | |||
| S Jun 18, 2021 at 2:39 | |||||
| Jun 17, 2021 at 9:43 | answer | added | fosco | timeline score: 6 | |
| Jun 17, 2021 at 6:27 | comment | added | David Roberts♦ |
Some richer notion of natural might be needed, of course, such as dinaturality, but I haven't thought too hard about it. One issue is that there is covariant and contravariant behaviour in the middle slot (b in your notation)
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| Jun 17, 2021 at 6:17 | comment | added | David Roberts♦ | As I'm sure you're aware, the pointwise evaluation map $ev\colon B^A \times A \to B$ has a universal property, but that doesn't quite help here! The trick is to identify what data in the map is variable, and which is fixed. Or, to put it another way, what variable is the map natural in? For $ev$, we fix $A$ and $B$ and let the thing in the $B^A$ slot vary, to consider generic $X\times A\to B$. Then $B^A$ together with $ev$ is the universal thing. What is the analogue for your setup? | |
| Jun 17, 2021 at 5:04 | comment | added | Johan Thiborg-Ericson | Anyways, I'll add a reference to your proposed theorem, and leave it to others to decide if my cartesian closed, non-2-category setting is enough to motivate a separate question. My main defence is that my question has more of a popular-science nature that might attract the attention of computer scientists, but I am unsure if that is an asset or a liability… | |
| Jun 17, 2021 at 4:52 | comment | added | Johan Thiborg-Ericson | @fosco: I decided to post this question because it seemed to me your question and proposed theorem was about something else that I didn't quite understand (I am but a mere computer scientist), but I will try to read it again more carefully, and update my question with a reference to your proposed solution. At a first glance, the main difference is the setting. You assume a 2-Category, but as a computer scientist, most of the time I only have the benefit of working in a category that is cartesian closed (Set). But then again, your theorem should be applicable to monads, I think, and that's good | |
| Jun 16, 2021 at 20:56 | comment | added | fosco | (I think my question is $\epsilon$-near to yours; if you have more doubts, feel free to ask) | |
| Jun 16, 2021 at 20:35 | comment | added | fosco | This question is not uninteresting, not off topic, and not too broad; it is in fact a very interesting one, and I asked something similar quite some time ago: mathoverflow.net/questions/315211/… | |
| Jun 16, 2021 at 18:05 | comment | added | Paul Taylor | I know the Thought Police are going to come along and Close this question. It's very difficult to think of a meaningful positive answer and indeed I've already thought of two situations where there is a choice of ways to compose. However, I believe that the question should stay here as a philosophical challenge to categorists to find positive answers. | |
| Jun 16, 2021 at 16:26 | review | First posts | |||
| Jun 16, 2021 at 16:43 | |||||
| Jun 16, 2021 at 16:24 | history | asked | Johan Thiborg-Ericson | CC BY-SA 4.0 |