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the page action is also a mess. I have added a pointer to the somewhat more comprehensive module and am hereby moving the following discussion box from there to here:
[ begin forwarded discussion ]
+–{.query} I am wondering if we will need the notion of action which works in categories with product, i.e. $G\times X\to X$ and so on. There is also an action of one Lie algebra on another (for instance in some definitions of crossed module of Lie algebra, where $Aut$ is replaced by the Lie algebra of derivations. (a similar situation would seem to exist in various other categories where action is needed in a slightly wider context. I think most would be covered by an enriched setting but I am not sure.) Thoughts please.Tim
Yes, I think certainly all those types of action should eventually be described somewhere, possibly on this page. -Mike
Tim: I have added some of this above. There should be mention of actions of a monoid in a monoidal category on other objects, perhaps.
Mac Lane, VII.4, only requires a monoidal category to define actions. – Uday =–
[ end forwarded discussion ]
Added a pointer to actegory under the section on actions of a category.
Added to action a footnote. Think there are good reasons to doing this, and doing it this way. Would expand upon request.
I suppose.
I am just noticing now footnote 1 on that page, though, which I don’t like. On one hand I think it is at best misleading: I don’t know what is meant by “the product of a monoid and a set doesn’t commute”; I would say that the reason left and right actions of a monoid are different is that the monoid may not be commutative. And the comment about directed graphs, while perhaps interesting, is totally unrelated and thus doesn’t belong in the same footnote. Furthermore, I don’t think this material should be in a footnote at all. If no one objects I may try to improve it.
Mike, many thanks for the comment. I would like to see them improved. No time to expand now.
Here the total space $Y/X$ of this bundle is typically the “weak” quotient (for instance: homotopy quotient) of the action, whence the notation…For more on this perspective on actions see at ∞-action.
That final link employs the $Y \sslash X$ notation, so it seems odd to have drawn attention to the notation of $Y/X$ as being typically weak.
Do we have a fixed convention for quotient notation?
Do we have a fixed convention for quotient notation?
I am afraid we don’t. The general nPOV would suggest that by default a single slash means the homotopy quotient, but in some cases that just seems too dangerous to leave implicit.
I removed the former footnote 1 at action, incorporating its content (clarified) into the text in appropriate places.
Many thanks Mike for cleaning this up.
I’ve expanded the actions_of_a_set section to include the notion of free category action and some other additions. Of course with my abilities things may need fixing.
Mike had added
(This is a sort of “Grothendieck construction”.)
which I totally don’t understand in this context.
Is this somehow related to a category of quivers with edges labeled by $L$ is the slice category $Quiv/Rose(L)$ where $Rose(L)$ is the one object quiver with $L$ edges?
(that category has morphisms that preserve and reflect edge labels. I can think of other morphisms that involve a change in labels.)
Suppose $Q$ is a general quiver, and define a (set-)representation of $Q$ to be a quiver map $Y:Q\to Set$. Thus every object of $Q$ is assigned a set $Y_x$ and every arrow $f:x\to y$ is assigned a function $Y_f : Y_x \to Y_y$. Now write down the usual definition of the Grothendieck construction as if $Q$ were a category and $Y$ a functor. Since $Q$ isn’t a category, the result won’t be a category either, but it will be a quiver with a map to $Q$. If $Q = Rose(X)$, then a representation of $Q$ is just an action of $X$ on a set $Y$, and this reproduces the quiver described in the entry, with the map to $Q$ assigning the labels as you suggest. (Is “$Rose(X)$” standard notation in quiver-theory?)
It would be reasonable to put this on the lab somewhere, but I’m not sure where.
Would there be discrete fibration-like lifting conditions on the map of quivers? The definition doesn’t require the composition operation.
Re
Is “$Rose(X)$” standard notation in quiver-theory?
from 10: not to my knowledge. Absence is hard to prove, yet I think there simply does not exist any usual term for this. In undirected contexts, people talk about bouquets a lot (flowery that, too), but in the directed setting: no, I think not.
(Small terminological comment on
Is this somehow related to a category of quivers with edges labeled by $L$ is the slice category $Quiv/Rose(L)$ where $Rose(L)$ is the one object quiver with $L$ edges?
from 9: in my opinion, the only sensible term for “the one object quiver with $L$ edges” is
which is also attested here and there on the web. Using “one object” seems misleading, or at least wrong emphasis, to me, for known reasons: a quiver is not a category; each quiver consists of sets, and calling sets objects is not wrong but can be wrong emphasis. )
David: yes, it seems that way.
That’s a good point to highlight. I have expanded this as follows:
Group actions, especially continuous actions on topological spaces, are also known as transformation groups (e.g. Bredon 72, tom Dieck 79, tom Dieck 87). Alternatively, if the group $G$ that acts is understood, one calls (Bredon 72, Ch. II) the space $X$ equipped with an action by $G$ a topological G-space (or G-set, G-manifold, etc., as the case may be).
and added pointer to these references.
Maybe there is a historical comment in order, as these days “transformation groups” seems not be used at all anymore?
As an aside, I find there is a missed opportunity here in fixing up the terminology:
After all, “group” is a somewhat unfortunate abbreviation of “group of symmetries” (unfortunate because it dropped the essential meaningful aspect while keeping an ad hoc technical term) and that in turn originates in “groups of symmetry transformations”, as originally (i.e. historically) all “groups” were really understood through/as their group actions, in the first place. So in speaking of “transformation groups” one is going back to the roots of the subject while insisting on historically evolved twists of what would have been its natural terminology.
Imagine in a bookshelf seeing the title “Symmetry transformations” – everyone knows at once what it’s about. Instead we have “Group theory” which (if you manage to remember what this looks like to the non-expert eye) is so unevocative of its subject matter that even modern search engines keep mixing it up with “Group therapy” – and that’s well-deserved, if overly mild, punishment for being so careless about the naming of concepts so profound.
Group actions… are also known as transformation groups …
Speaking of a situation
$\alpha: G \times X \to X,$I would call $\alpha$ the group action, and $G$ the transformation group.
You can see how the shift to plain “group” took place as one looked to study such $G$ without reference to any specific $X$.
It would be interesting to try to study groups only via their representations. Take simple groups for example. Can one recognise a simple group through its representation theory? They have no 1-dimensional representations over $\mathbb{C}$, but that is not a characterisation. However, can one for example say anything interesting about those groups with no 1-dimensional complex representations as a class?
For something easier, one can recognise abelian groups by their representations: they are exactly those groups with only 1-dimensional (irreducible) representations. Thus one does kind of see simpleness roughly reflected in the 1-dimensional representation theory: maximally non-simple groups (i.e. abelian groups) have only 1-dimensional representations, maximally simple groups have none.
Simple groups have no 2-dimensional representations either, but I’m not sure if groups with only 2-dimensional representations are interesting.
(Certain simple groups have a 3-dimensional representation.)
Actually I think that finite groups with only 1- or 2-dimensional irreducible complex representations are exactly what one would naively guess: those groups with an abelian normal subgroup of index 2. So everything works out quite nicely up to dimension 2. I’m not sure if one will be able to say anything in dimension 3 though.
Hmm, actually it makes a big difference to require to not have any representations in dimension higher than 3, this will exclude the simple groups I think, and maybe the naive guess (abelian normal subgroup of index 2 or 3) will work. Maybe this does generalise to all dimensions (i.e. one asks for abelian normal subgroups of indices corresponding to the prime decomposition).
22, 23, 24 While I strongly agree with sensitive discussion in 23, I still think that the summary in 24 is missing parts of the point from the mathematical practice, even though major authorities quoted in 22 have overdefined in this vain. A modern practical mathematician will indeed tend to call transformation group a group with some standard (or often even defining) action of it, for example the diffeomorphism group of a manifold, or permutation group on $n$ letters. On the other hand, when one is fixing a group and considering varying actions, “realizations”, faithful or not, nowdays one would like to call it as action or realizations, or (nonlinear) representations, rather than varying transformation groups.
25 While doing group theory abstractly without their actions is loosing the real power of the theory (as Miles Reid used to criticise), the reducing point of view which I heard from a professor as a junior student that a group is interesting only when it is realized as a linear representation is also at lest demotivating (for me, that blatantly reductionistic point of view (“group makes sense only if realized as matrices”) has pushed me away psychologically from representation theory for my entire career). For example, combinatorial group theory with free groups and their subgroups has its own content which start with combinatorics of generators and relations, and then introduces connections to topology, automorphisms etc. (mainly non-linear). I knew the beauty of combinatorial group theory from my hi school having read Magus’s children book on groups and their graphs, and then as a freshman first pages of deep book by Lindon and Schupp. Then martyrs of the law nothing makes sense if it is not a concrete matrix killed my love for group theory for many years. Mathematics has many facets and insisting that only one point of view is right is doing more harm than good. Of course, the questions in 25-29 are interesting.
on the usage of the term “transformation group” for “group action” I have added pointer to Section 1 of:
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872) Mathematische Annalen volume 43, pages 63–100 1893 (doi:10.1007/BF01446615)
English translation by M. W. Haskell:
A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (euclid:1183407629 KleinRetyped.pdf:file)
added also pointer to:
Thanks! I didn’t know that this is on the arXiv. Also that retyped pdf was uploaded many years ago, I only copied it over here now form Erlangen program. In any case, thanks for giving the arXiv link, that indeed makes the other pdf redundant.
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