Timeline for Can such categorical notion of action be formalized?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 5, 2015 at 5:01 | comment | added | მამუკა ჯიბლაძე | There is a standard trick for that too, - actually I mentioned it here in an answer: internal abelian group $p:A'\to A$ in algebras over an algebra $A$ splits as a semidirect product of $A$ and the kernel of $p$, and can be reconstructed from $A$ and the action on the kernel. The notion of semidirect product and action depends on the variety where the algebra resides. This actually to a certain extent works even for monoids, by calling "kernel" the family of inverse images of elements of $A$ (in better cases determined by each other). | |
| Sep 4, 2015 at 17:56 | comment | added | sure | If you refer to the action on itself, I agree. It still seems that given a structure with some operations, and another structure with other operations, it is enough to know the nature of the operations to define the good concept of action. I guess one would have to do some kind of "zoology" of operations (are they associative? considered as additive or multiplicative? distributive?) in order to define such construction and look for a universal way to define them. The concept of action is unifying, so there must be something to understand there. | |
| Sep 4, 2015 at 17:43 | comment | added | მამუკა ჯიბლაძე | Well still there is some variation. For example, just compare a Lie algebra action and an associative algebra action. There must be some intrinsic explanation why the "correct" action is by derivations in the first case but not in the second, right? | |
| Sep 4, 2015 at 15:58 | comment | added | sure | @მამუკაჯიბლაძე: I find it weird that they formalize it that way. A monoid set-action is easily defined through the usual concept of map $.: M\times X \rightarrow X$ that is associative and stuff. Moreover, it seems that representation of groups or lie algebra are just actions in the same sense (instead of acting on set, they act on a vector space $V$, $.: G\times V \rightarrow V$). I'll try to think if there exists an universal way using this "sketch" approach. Thanks for the references though. | |
| Sep 2, 2015 at 14:31 | comment | added | მამუკა ჯიბლაძე | In few words - if you are in a variety where kernels make sense (so e. g. monoids are ruled out) then you look at retracts $X\rightarrowtail Y\twoheadrightarrow X=1_X$, and find out how $X$ "acts" on the kernel of $Y\twoheadrightarrow X$. | |
| Sep 2, 2015 at 14:27 | comment | added | მამუკა ჯიბლაძე | Probably the best place to start is "Actors in categories of interest" as it contains sort of brief survey, works in sufficient but not too overwhelming generality, and contains lots of both historical and relatively fresh references | |
| Sep 2, 2015 at 14:22 | comment | added | sure | @მამუკაჯიბლაძე: do you have any references where this concept is introduced? I can't find it easily using google and arxiv. Thanks. | |
| Sep 2, 2015 at 13:02 | comment | added | მამუკა ჯიბლაძე | Actually to come up with the correct notion of action is not entirely trivial. There is something called actor or split extension classifier which gives endomorphisms for monoids, automorphisms for groups, derivations for Lie algebras, bimultiplications for associative algebras, biderivations for Leibniz algebras... | |
| Sep 2, 2015 at 12:28 | comment | added | Zhen Lin | It is not as general as you ask for. I am just pointing out that there is something that captures the examples you mention. | |
| Sep 2, 2015 at 12:14 | comment | added | sure | What I want is to take two categories and consider that the resulting action of one category on the other is another category as I said. I don't see how what you say is related to what I want, mind you expliciting? | |
| Sep 2, 2015 at 11:29 | comment | added | Zhen Lin | These are all essentially examples of monoid actions for a monoid in a monoidal category. | |
| Sep 2, 2015 at 9:15 | history | asked | sure | CC BY-SA 3.0 |