Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: deprecated: mathematics

Topic: magmas and set actions

David Michael Roberts (Aug 29 2023 at 03:21):

A shower-type thought that I wrote down a while ago: a magma is nothing but the action of a set on itself. Since the set itself has no operations or additional data, there are no compatibility axioms. More generally, a left action of a set $X$ on a set $Y$ is an arbitrary function $a\colon X\times Y\to Y$. An "equivariant" map $f\colon Y_1\to Y_2$ of "$X$-sets" is a function such that $a_2(x,f(y)) = f(a_1(x,y))$ for all $x\in X,\ y\in Y_1$ (using $a_i$ for the action on $Y_i$).

I just found it amusing.

David Michael Roberts (Aug 29 2023 at 03:24):

One could also define an $X$-action on $Y$ to be an arbitrary function $X\to Y^Y$, so one can see that it is an $X$-indexed family of endomorphisms of $Y$, satisfying no axioms.

David Michael Roberts (Aug 29 2023 at 03:24):

Generalisation to other categories than $\mathbf{Set}$ are immediate, and an exercise for the reader.... ;-)

Mike Shulman (Aug 29 2023 at 03:25):

I guess a place where this comes up in "real life" is that a [[list object]] $L A$ is the free $A$-set on a point.

David Michael Roberts (Aug 29 2023 at 05:10):

Hmm, it's close, if not exactly that. You have the additional relation of associativity in the list monoid. However, I guess considering only the left $A$-action, you only get the arrangement of parentheses that are all associated in one way (unlike thinking about the actual underlying magma, where one can get arbitrary matched-parentheses arrangement). So probably you are right, since the order of concatenation is unambiguous...

John Baez (Aug 29 2023 at 08:04):

I used actions of sets on objects in an arbitrary category in my paper with @Brandon Coya and Franciscus Rebro, Props in network theory.

We say a set L acts on an object x of some category if that object is equipped with a function L $\to$ hom(x,x). So this is compatible with @David Michael Roberts idea: a set with an action on itself is the same as a magma.

We showed that circuits with wires labeled by elements of some set L are morphisms in the prop for commutative special Frobenius monoids whose underlying object has an L-action.

John Baez (Aug 29 2023 at 08:08):

This is a rephrasing of a result due to Rosebrugh, Sabadini and Walters. But we wanted to think of circuits of different kinds as morphisms in different props.

John Baez (Aug 29 2023 at 08:10):

You can also say an object x in a closed category acts on an object y if you've got a morphism x $\to$ [y,y], where [-,-] is the internal hom.

John Baez (Aug 29 2023 at 08:11):

Of course for a monoidal closed category this is the same as a morphism x $\otimes$ y $\to$ y.

John Baez (Aug 29 2023 at 08:12):

It's fun to read Schafer's book Nonassociative Algebras while keeping this stuff in mind. A nonassociative (not necessarily unital) algebra is a vector space acting on itself.

John Baez (Aug 29 2023 at 08:13):

What's great about Schafer's book is how many of the concepts and results about associative algebras generalize to the nonassociative case!

Mike Shulman (Aug 29 2023 at 14:24):

Hmm, apparently someone on the nLab decided that "list object" should redirect to "free monoid", which I think is wrong. A list object is often also a free monoid, but the definition of a list object is that it's the free $A$-set on a point: it's the initial object of the category of objects $X$ equipped with morphisms $\mathrm{nil} : 1\to X$ and $\mathrm{cons} : A \times X \to X$.

Kevin Arlin (Aug 29 2023 at 16:10):

I personally hadn't registered the difference until a recent conversation on here, and I bet that's what happened with whoever instantiated that link.

David Michael Roberts (Aug 29 2023 at 21:54):

@Mike Shulman is the presence of nil just to ensure A-sets are inhabited? Or is that preserved structure? I get that if one allows empty A-sets, then the empty one would be initial instead.

David Michael Roberts (Aug 29 2023 at 21:55):

Oh, wait, you said free on a point, I see.