Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorRodMcGuire
   • CommentTimeDec 22nd 2014
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI've added the below to the Idea section of [[action#idea]] as a simpler intro before jumping into delooping. Maybe some of the text in the footnote should be incorporated into the body, and I haven't changed anything that follows to jibe with it. > The simplest notion of action involves one set, $X$, acting on another $Y$ as a the function $act\colon X \times Y \to Y$. This can be [[currying|curried]] as $ \hat{act}\colon Y \to Y ^ X $ where $Y ^ X$ is the (monoidal) [[function set|set of functions]] from $X$ to $Y$.[^SetActions] [^SetActions]: In the category [[Set]] there is no difference between the above left action and the right action $actR\colon Y \times X \to Y$ because the product commutes. However for the action of a [[monoid]] on a set (sometimes called [[MSet| M-set]] or M-act) the product of a monoid and a set does not commute so the left and right actions are different. The action of a set on a set is the same as an arrow labeled [[directed graph]] $arrows\colon vertices \times labels \to vertices$ which specifies that each vertex must have a set of arrows leaving it with one arrow per label, and is also the same as a simple (non halting) [[deterministic automaton]] $transition\colon inputs \times states \to states$. > Generalized notions of action use entities from categories other than $Set$ and involve an [[exponential object]] such as $Y ^ X$.

   I’ve added the below to the Idea section of action#idea as a simpler intro before jumping into delooping. Maybe some of the text in the footnote should be incorporated into the body, and I haven’t changed anything that follows to jibe with it.

   The simplest notion of action involves one set, $X$, acting on another $Y$ as a the function $act\colon X \times Y \to Y$. This can be curried as $\hat{act}\colon Y \to Y ^ X$ where $Y ^ X$ is the (monoidal) set of functions from $X$ to $Y$.¹

   Generalized notions of action use entities from categories other than $Set$ and involve an exponential object such as $Y ^ X$.

   ---

   1. In the category Set there is no difference between the above left action and the right action $actR\colon Y \times X \to Y$ because the product commutes. However for the action of a monoid on a set (sometimes called M-set or M-act) the product of a monoid and a set does not commute so the left and right actions are different. The action of a set on a set is the same as an arrow labeled directed graph $arrows\colon vertices \times labels \to vertices$ which specifies that each vertex must have a set of arrows leaving it with one arrow per label, and is also the same as a simple (non halting) deterministic automaton $transition\colon inputs \times states \to states$. ↩

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 23rd 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDid you mean to curry that way? I'm more used to seeing $X \to Y^Y$, where it's $Y^Y$ that carries a monoid structure. For example, an action $M \times Y \to Y$ of a monoid $M$ is is tantamount to a homomorphism of monoids $M \to Y^Y$.

   Did you mean to curry that way? I’m more used to seeing $X \to Y^Y$, where it’s $Y^Y$ that carries a monoid structure. For example, an action $M \times Y \to Y$ of a monoid $M$ is is tantamount to a homomorphism of monoids $M \to Y^Y$.

3. • CommentRowNumber3.
   • CommentAuthorRodMcGuire
   • CommentTimeDec 23rd 2014
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItex> Did you mean to curry that way? Whoops. To write that I was mainly copying and pasting, and I also have something maybe called "order dyslexia". (correctly taking down phone numbers is a pain) I'll wait a little (to maybe tomorrow) to see if there are any more comments or whether someone else jumps in to rewrite the intro, before I go change things..

   Did you mean to curry that way?

   Whoops. To write that I was mainly copying and pasting, and I also have something maybe called “order dyslexia”. (correctly taking down phone numbers is a pain)

   I’ll wait a little (to maybe tomorrow) to see if there are any more comments or whether someone else jumps in to rewrite the intro, before I go change things..

4. • CommentRowNumber4.
   • CommentAuthorRodMcGuire
   • CommentTimeDec 23rd 2014
   • (edited Dec 23rd 2014)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI've fixed my addition now to the below. Have I again made mistakes? >The simplest notion of an action involves one set, $X$, acting on another $Y$ which is defined as a function $act\colon X \times Y \to Y$. This can be [[currying|curried]] to $\hat{act}\colon X \to Y ^ Y $ where $Y ^ Y$ is the (monoidal) [[function set|set of functions]] from $Y$ to $Y$, also called the [[endofunction]] $End(Y)$.[^SetActions] [^SetActions]: In the category [[Set]] there is no difference between the above left action and the right action $actR\colon Y \times X \to Y$ because the product commutes. However for the action of a [[monoid]] on a set (sometimes called [[MSet| M-set]] or M-act) the product of a monoid and a set does not commute so the left and right actions are different. The action of a set on a set is the same as an arrow labeled [[directed graph]] $arrows\colon vertices \times labels \to vertices$ which specifies that each vertex must have a set of arrows leaving it with one arrow per label, and is also the same as a simple (non halting) [[deterministic automaton]] $transition\colon inputs \times states \to states$. >Generalized notions of action use entities from categories other than $Set$ and involve an [[endofunctor]] [[exponential object]] such as $Y ^ Y$.

   I’ve fixed my addition now to the below. Have I again made mistakes?

   The simplest notion of an action involves one set, $X$, acting on another $Y$ which is defined as a function $act\colon X \times Y \to Y$. This can be curried to $\hat{act}\colon X \to Y ^ Y$ where $Y ^ Y$ is the (monoidal) set of functions from $Y$ to $Y$, also called the endofunction $End(Y)$.¹

   Generalized notions of action use entities from categories other than $Set$ and involve an endofunctor exponential object such as $Y ^ Y$.

   ---

   1. In the category Set there is no difference between the above left action and the right action $actR\colon Y \times X \to Y$ because the product commutes. However for the action of a monoid on a set (sometimes called M-set or M-act) the product of a monoid and a set does not commute so the left and right actions are different. The action of a set on a set is the same as an arrow labeled directed graph $arrows\colon vertices \times labels \to vertices$ which specifies that each vertex must have a set of arrows leaving it with one arrow per label, and is also the same as a simple (non halting) deterministic automaton $transition\colon inputs \times states \to states$. ↩

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for looking into improving the Idea-section of this entry! It badly deserved to be expanded to something less stubby. (There are many old $n$Lab entries with very stubby idea-sections. Would be nice if eventually we'd come back to all of them and improve them.) While I like the idea of starting the explanation with the action of a set on another set, this runs a bit counter to what the Idea-section said (and says) a bit further below, namely that the key aspect of an action is that it is an algebraic thing that acts. This is precisely where those functors on deloopings come in, which were the only thing that the text had previously mentioned. So I have tried to add a bit of glue that connects your addition to the little bit that the previous text tried to say. In the course of this I ended up writing and re-writing a few more paragraphs of the Idea-section. Please everybody check it out [here](http://ncatlab.org/nlab/show/action#Idea) and feel invited to further expand and fine-tune. (And please feel invited to complain and roll back if you are unhappy with what I did.)

   Thanks for looking into improving the Idea-section of this entry! It badly deserved to be expanded to something less stubby.

   (There are many old $n$Lab entries with very stubby idea-sections. Would be nice if eventually we’d come back to all of them and improve them.)

   While I like the idea of starting the explanation with the action of a set on another set, this runs a bit counter to what the Idea-section said (and says) a bit further below, namely that the key aspect of an action is that it is an algebraic thing that acts. This is precisely where those functors on deloopings come in, which were the only thing that the text had previously mentioned.

   So I have tried to add a bit of glue that connects your addition to the little bit that the previous text tried to say.

   In the course of this I ended up writing and re-writing a few more paragraphs of the Idea-section.

   Please everybody check it out here and feel invited to further expand and fine-tune.

   (And please feel invited to complain and roll back if you are unhappy with what I did.)

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 24th 2014
   • (edited Dec 24th 2014)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexSome comments: (1) the word "curried" is somewhat amusing, as obviously we're dealing with a back-formation from the last name Curry; personally I've been writing "curryed" although I concede it also looks funny (and I wouldn't insist on it); (2) I wouldn't say "monoidal" since the only typical usage of that word is from "monoidal category" which is not really relevant at this point -- I think it would be cleaner to say "where $Y^Y$ is the set of functions from $Y$ to $Y$, forming the endofunction monoid $End(Y)$", (3) you might say in the following sentence "generalized notions of actions may be formulated in monoidal categories other than $Set$, with the possibility of currying available if the monoidal category is also closed (allowing one to form endofunction exponential objects $Y \multimap Y$)". (4) I'm not sure what you're driving at in the footnote. The cartesian product is symmetric whether we speak of two sets or a monoid and a set. But the notion of a monoid action on a set involves some more axioms on top of the bare structure of the action, and it is true that left actions of a monoid $M$ which are tantamount to monoid morphisms $M \to Y^Y$ are clearly distinguished from right actions which amount to monoid morphisms $M^{op} \to Y^Y$, with the distinction vanishing if $M$ is a commutative monoid. (I was writing this before I saw Urs's comment; it might be irrelevant now.)

   Some comments: (1) the word “curried” is somewhat amusing, as obviously we’re dealing with a back-formation from the last name Curry; personally I’ve been writing “curryed” although I concede it also looks funny (and I wouldn’t insist on it); (2) I wouldn’t say “monoidal” since the only typical usage of that word is from “monoidal category” which is not really relevant at this point – I think it would be cleaner to say “where $Y^Y$ is the set of functions from $Y$ to $Y$, forming the endofunction monoid $End(Y)$”, (3) you might say in the following sentence “generalized notions of actions may be formulated in monoidal categories other than $Set$, with the possibility of currying available if the monoidal category is also closed (allowing one to form endofunction exponential objects $Y \multimap Y$)”. (4) I’m not sure what you’re driving at in the footnote. The cartesian product is symmetric whether we speak of two sets or a monoid and a set. But the notion of a monoid action on a set involves some more axioms on top of the bare structure of the action, and it is true that left actions of a monoid $M$ which are tantamount to monoid morphisms $M \to Y^Y$ are clearly distinguished from right actions which amount to monoid morphisms $M^{op} \to Y^Y$, with the distinction vanishing if $M$ is a commutative monoid.

   (I was writing this before I saw Urs’s comment; it might be irrelevant now.)

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeDec 27th 2014
   • (edited Dec 27th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy coincidence I just see that a related issue appears in the old entry _[[faithful representation]]_. This used to speak of the "adjoint map" without saying what is meant. I have slightly expanded the Definition-section now to make this clearer. Please don't hesitate if you feel like editing this entry further, it would deserve it.

   By coincidence I just see that a related issue appears in the old entry faithful representation.

   This used to speak of the “adjoint map” without saying what is meant. I have slightly expanded the Definition-section now to make this clearer. Please don’t hesitate if you feel like editing this entry further, it would deserve it.