Timeline for answer to Why is the category of fields seemingly so poorly behaved? by Zhen Lin
Current License: CC BY-SA 4.0
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13 events
| when toggle format | what | by | license | comment | |
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| Jul 26, 2020 at 4:51 | audit | First posts | |||
| Jul 26, 2020 at 4:51 | |||||
| Jul 22, 2020 at 16:30 | audit | First posts | |||
| Jul 22, 2020 at 16:30 | |||||
| Jul 17, 2020 at 9:20 | audit | First posts | |||
| Jul 17, 2020 at 9:21 | |||||
| Jul 15, 2020 at 4:44 | comment | added | Kevin Carlson | @Brahadeesh You can find this theorem, and a number of related ones, in Adamek-Rosicky-Vitale's book on algebraic theories. | |
| Jul 15, 2020 at 1:42 | comment | added | Zhen Lin | There are many (interrelated) reasons for excluding the zero ring. From an algebraic geometry perspective: fields are points, but the zero ring is just empty; from the algebra perspective: the quotient of a commutative ring by an ideal is a field if and only if the ideal is maximal, etc. | |
| Jul 14, 2020 at 13:59 | comment | added | tomasz | Does the inequality $0\neq 1$ matter so much? Adding the zero ring to the category of fields does not seem to alter the landscape much, since it would be an isolated object anyway. | |
| Jul 14, 2020 at 13:30 | comment | added | Zhen Lin | Products are a particular kind of limit. Free objects are what you get by applying the left adjoint of the forgetful functor. It is difficult to reason about counterfactuals but since the category of fields is a finitely accessible category and the forgetful functor is finitely accessible, you could say that if the category had limits then everything would have worked. | |
| Jul 14, 2020 at 13:15 | comment | added | Ducky | Thanks for the helpful answer! So then the failure of products, free fields, etc. can all be seen as a failure for certain diagrams to have limits? | |
| Jul 14, 2020 at 13:00 | comment | added | Zhen Lin | I no longer remember where it is written in the literature. The "only if" direction is easy, at any rate; the "if" direction requires more work but is easy if you know about Lawvere theories, locally presentable categories, etc. | |
| Jul 14, 2020 at 12:22 | comment | added | user279515 | Could you also add a source for the complete theorem that you mention at the start? Thanks! | |
| Jul 14, 2020 at 11:33 | comment | added | Zhen Lin | @Asaf For some reason I seem to have too much free time these last few months... | |
| Jul 14, 2020 at 9:48 | comment | added | Asaf Karagila♦ | Fancy seeing you here! | |
| Jul 14, 2020 at 2:10 | history | answered | Zhen Lin | CC BY-SA 4.0 |