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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.
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