Timeline for Is there a general notion of semigroup action?
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 22, 2015 at 20:36 | vote | accept | Thomas Klimpel | ||
| Sep 1, 2014 at 17:23 | answer | added | Robin Cockett | timeline score: 3 | |
| Sep 1, 2014 at 11:49 | comment | added | Andrej Bauer | ResearchGate is evil. Here is a link to Funk & Hofstra's original article: tac.mta.ca/tac/volumes/24/6/24-06abs.html Any yes, that's an excellent reference! | |
| Sep 1, 2014 at 10:07 | comment | added | მამუკა ჯიბლაძე | @BenjaminSteinberg Thank you for the clarification. I've found the paper Topos theoretic aspects of semigroup actions by Funk and Hofstra which enlightened me :) It seems to be much more complicated and interesting than I thought. And I believe it is relevant for the question. | |
| Sep 1, 2014 at 9:58 | comment | added | Thomas Klimpel | @AndrejBauer My motivation is more the question: "What would be a possible definition of inverse semigroup action that plays nice with the representation theorem." I tried to use category theory to get around my problems with partial functions, but didn't come far. One issue is that "my inverse semigroup and the object live in the same category". The other issue is that I need to read more about "restriction categories", and maybe also continue reading more of the "established theory of inverse semigroups". | |
| Sep 1, 2014 at 7:53 | comment | added | Andrej Bauer | I summon Robin Cockett! | |
| Sep 1, 2014 at 7:38 | comment | added | Andrej Bauer | So, is your question: "What is the categorical generalization of the theorem which says that every inverse semigroup can be repersented by partial symmetries?" The first part of your new main question is easily answered: yes, it makes sense because every inverse semigroup is a semigroup. For the second question, I need to recall how the partial symmetries thing works... | |
| Aug 31, 2014 at 23:53 | comment | added | Thomas Klimpel | @AndrejBauer "Let's continue the discussion." In case this was directed at me and you are still interested in "the discussion", I would have done most of "my homework" now. I have read enough to make sense of nearly everything in the comments here. I have tried to distill my concerns about the proposed "solution" into questions that I "hope" can be answered. I also provided an answer myself, which shows in painstaking detail how the proposed "solution" is insufficient to achieve the intended goal (and has additional "undesired" implication if followed consequently). | |
| Aug 31, 2014 at 13:24 | answer | added | Thomas Klimpel | timeline score: 1 | |
| Aug 31, 2014 at 12:51 | history | edited | Thomas Klimpel | CC BY-SA 3.0 |
Tried again to clarify my main question, and how/why the side issues affect my solution attempts
|
| Aug 30, 2014 at 14:03 | answer | added | Benjamin Steinberg | timeline score: 4 | |
| Aug 30, 2014 at 13:49 | comment | added | Benjamin Steinberg | The idempotent splitting of an inverse semigroup is not a groupoid. | |
| Aug 30, 2014 at 12:53 | comment | added | Andrej Bauer | I am in fact downvoting the question because it is unclear what it is asking. Let's continue the discussion. | |
| Aug 30, 2014 at 12:19 | comment | added | Zhen Lin | @QiaochuYuan It's a nice trick for $\mathbf{Set}$ but it doesn't work properly in general: you would need every subobject to have a complement in the strong sense. | |
| Aug 30, 2014 at 10:24 | comment | added | მამუკა ჯიბლაძე | As @Andrej, I do not understand well what exactly you want to achieve but I have a vague feeling nonetheless that idempotent splitting may be used to unify everything in sight. It is always the case that the category of actions of a monoid $M$ on objects of an idempotent-split category $C$ is equivalent to the category of actions of the category of idempotents of $M$ on objects of $C$. I believe both semigroups and inverse semigroups are incorporated as particular cases of this. If I am not mistaken, inverse semigroups are precisely semigroups whose category of idempotents is a groupoid. | |
| Aug 30, 2014 at 9:39 | history | edited | Thomas Klimpel | CC BY-SA 3.0 |
Tried to me more explicit about my problems with respect to "how to make it work"
|
| Aug 30, 2014 at 9:07 | comment | added | Thomas Klimpel | @AndrejBauer "..., so I don't really understand what you'd like." I want to understand how to make it work, including inverse semigroups and groupoids. As a first step, I try to get "implicit" agreement that the definition of groupoid action (and semigroupoid action/category action) is fine and "obvious". The challenging part is to see how inverse semigroups can be taken care of appropriately in the definition of semigroup action. One potential solution could also be to declare that an "inverse semigroups action" need not be a special case of a "semigroup action". | |
| Aug 30, 2014 at 8:48 | comment | added | Andrej Bauer | Your question about getting a notion of partial morphism is separate from what makes a semigroup action. For that, I recommed reading: J. Robin B. Cockett, Stephen Lack: Restriction categories I: categories of partial maps. Theor. Comput. Sci. 270(1-2): 223-259 (2002), and also the second paper J. Robin B. Cockett, Stephen Lack: Restriction categories II: partial map classification. Theor. Comput. Sci. 294(1/2): 61-102 (2003) | |
| Aug 30, 2014 at 8:46 | comment | added | Andrej Bauer | The action of a semigroup $G$ in a category $\mathcal{C}$ is given by a functor $G \to \mathcal{C}$, where we view $G$ as a category with a single object, and the morphisms are the elements of $G$. This comes down to: an action is given by an object $X$ in $\mathcal{C}$ and a semigroup homomorphism $G \to \mathrm{Hom}(X,X)$. It's all completely analogous to group actions, so I don't really understand what you'd like. | |
| Aug 30, 2014 at 8:23 | history | edited | Thomas Klimpel | CC BY-SA 3.0 |
stupid me: a semigroupoid is just a category
|
| Aug 30, 2014 at 8:22 | comment | added | Qiaochu Yuan | If $C$ is any category with a terminal object $1$, a "canonical" candidate for the category of "partial morphisms" in $C$ is the category of pointed objects in $C$ (an object equipped with a map $1 \to c$). | |
| Aug 30, 2014 at 7:55 | history | asked | Thomas Klimpel | CC BY-SA 3.0 |