Timeline for $\mathcal A^{\mathcal A}\sim\mathcal B^{\mathcal B}\implies\mathcal A\sim\mathcal B\ ?$ (Does A^A ~ B^B imply A ~ B? --- A, B categories)

7 events

when toggle format	what		by	license	comment
Jun 13, 2019 at 18:59	vote	accept	Pierre-Yves Gaillard
Jun 13, 2019 at 18:58	history	edited	Matt Feller	CC BY-SA 4.0	Fixed End(G) and added a note at the end
Jun 13, 2019 at 18:51	comment	added	Matt Feller		Oops, yes! It should just be End(G). And you're right, the proof shows that the endofunctor categories are in fact isomorphic. I will edit my answer to emphasize that.
Jun 13, 2019 at 18:26	comment	added	Pierre-Yves Gaillard		Thanks! I don't doubt that your proof is correct but I'd like to make sure I understand it, and I'm very slow. Shouldn't $\text{End}(G,G)$ be $\text{End}(G)$? Also it seems to me that you found an example with $\mathcal A$ and $\mathcal B$ not equivalent but $\mathcal A^{\mathcal A}$ and $\mathcal B^{\mathcal B}$ isomorphic (not just equivalent). Am I right?
Jun 13, 2019 at 18:06	history	edited	Matt Feller	CC BY-SA 4.0	typo
Jun 13, 2019 at 17:26	history	edited	Matt Feller	CC BY-SA 4.0	added 76 characters in body
Jun 13, 2019 at 17:14	history	answered	Matt Feller	CC BY-SA 4.0