Timeline for Reference on inductive (direct) limit of ordered vector spaces and vector lattices
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2022 at 7:40 | vote | accept | erz | ||
| Apr 17, 2022 at 9:23 | answer | added | JHvdWalt | timeline score: 3 | |
| Feb 10, 2022 at 12:09 | comment | added | Jochen Wengenroth | Yes, that's right. It is however important to have a candidate for colimits, this helps a lot either to verify that the cadidate is (or rather, can be made) a colimt or that a colimit does not exist. | |
| Feb 10, 2022 at 0:22 | comment | added | erz | @JochenWengenroth but this does not prove existence of colimits, only that they are the way they are, right? | |
| Feb 9, 2022 at 7:55 | comment | added | Jochen Wengenroth | This does not exactly fulfil your request but nevertheless, I propose the following example: The forgetful functor $U:$ LCS $\to$ Vect has a right adjoint (endow a vector space with the trivial topology generated by the seminorm $0$). Therefore, $U$ preserves colimits. This implies and (perhaps even more importantly) explains that colimits in LCS are the colimits in Vect endowed with a suitable locally convex topology. If you consider instead the category of Hausdorff LCS, the forget functor does not have a right adjoint and colimits in that category are algebraically different. | |
| Feb 9, 2022 at 1:36 | comment | added | erz | @JochenWengenroth I seem to be able to prove the properties myself, I just don't want to dedicate a third of a short note that I am writing to developing this theory from scratch. Could you please give an example in a different category (locally convex spaces perhaps?) where the permanence properties are obtained for free? In my rudimentary understanding of category theory, I find it hard to see how the adjoint functor can "detect" the subcategories of the target category | |
| Feb 8, 2022 at 16:21 | comment | added | Jochen Wengenroth | If nobody knows a reference you might ask explicitely for the permanence properties. The construction of the inductive limit (category theorists nowadays seem to prefer the name colimit) involves the forgetful functor into the category of vector spaces. You might look for a right adjoint of that functor. This is sometimes useful to get permanence properties for free. | |
| Feb 8, 2022 at 11:00 | history | edited | erz | CC BY-SA 4.0 |
added 5 characters in body
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| Feb 8, 2022 at 2:38 | history | asked | erz | CC BY-SA 4.0 |