Timeline for Name for isomorphisms canonically identifying all elements in a category
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 29, 2023 at 9:20 | comment | added | Dominique Unruh | @JeremyRickard To be honest, I can't confirm or deny the latter since me and category theory have only a passing acquaintance. Personally, I'm often a bit wary of category-theory proofs because I can't see the way how to break them down to a concrete axiom system (e.g., ZFC, HOL, or whichever) but it might be my own limitation. | |
| Sep 28, 2023 at 13:20 | comment | added | Jeremy Rickard | Yes, fair enough. I was allowing myself enough “choice”. But I think only as much as is needed to prove that the two familiar definitions of an equivalence of categories (has an inverse up to natural isomorphism, and is fully faithful and essentially surjective) are the same? | |
| Sep 28, 2023 at 13:13 | comment | added | Dominique Unruh | @JeremyRickard It's not really relevant for my question, but about the question whether every category has such an isomorphism: I'm not convinced. The construction you mention would need the existence of a family $\phi_A$, which, I think, would need some strong form of the axiom of choice (over categories as opposed to over sets). So depending on the axioms you are willing to use, even the existence might be a problem. | |
| Sep 28, 2023 at 13:10 | comment | added | Dominique Unruh | @JeremyRickard Yes, I am trying to name the property. Basically, I would write a sentence such as "we show that $\iota_{AB}$ is XXX, that is satisfies ...". If there exists no term, that's also ok, then I'll make up something, but I wanted to use an existing term if there is one. | |
| Sep 27, 2023 at 16:43 | comment | added | Mike Shulman | I have never heard of a name for such a thing, but the first name that occurs to me is "wide clique". | |
| Sep 27, 2023 at 13:25 | comment | added | Sean Sanford | Upon choosing an object, this feels like it's equivalent to the data of a fixed adjoint equivalence to the trivial category. | |
| Sep 27, 2023 at 12:54 | comment | added | Jeremy Rickard | Ah, but reading the question more carefully, I think you’re asking for a name for the collection of isomorphisms, rather than a name for a category which has such a collection? | |
| Sep 27, 2023 at 12:49 | comment | added | Jeremy Rickard | Doesn’t every category in which all objects are isomorphic have this property? Fix an object $X$, and for each object $A$ pick an isomorphism $\varphi_A: X\to A$. Then let $\iota_{AB}=\varphi_B\varphi_A^{-1}$. | |
| Sep 27, 2023 at 11:27 | history | asked | Dominique Unruh | CC BY-SA 4.0 |