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Question about "dynamical systems," as Lawvere/Schnauel calls them in their baby book (ie digraph w exactly 1 arrow out of each point). What would a "chaotic" dynamical system be? In the book's second-edition appendices, they say the forgetful functor to Set has a right adjoint, associated with such a system, but I can't envision what such an object would be--what's "indiscreteness" in a deterministic graph? ...On that note, by the way, they also say that the left adjoint assigns the "free dynamical system" and say it has to do with natural numbers and recursion, but as far as I can see the left adjoint to the forgetful functor to set could only be the discrete functor, same as for any digraph.

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This question should probably include more details; I know what you're talking about because I happen to have read Lawvere and Schanuel, but to someone who hasn't (even someone who might otherwise be qualified to answer this question) this question is somewhat hard to understand. –  Qiaochu Yuan Sep 29 '12 at 21:09
Am I right to conclude that the category in question has maps $f\colon S\to S$ as objects and a morphisms from $f_1\colon S_1\to S_1$ to $f_2\colon S_2\to S_2$ is a map $\phi\colon S_1\to S_2$ with $f_2\circ \phi=\phi\circ f_1$? –  Hagen von Eitzen Sep 29 '12 at 21:12
Yes. That is the nature of the category. I can now see that it was inappropriate for me to "generate" the free object the same way I would have in Gph, simply because of the obvious embedding. I still don't quite get how the free functor generates the product with N, though. –  Adam Ray Sep 29 '12 at 22:15
@Adam: by "get" do you mean on an intuitive level or on the level of being able to write out a proof? The proof is a verification of the universal property. The intuitive level is to think of how you would write down a dynamical system which is as unconstrained as possible starting from a set $X$. Well, if you apply $f$ to each element of $X$, you'll get... a bunch of other things, $f(X)$, that you don't know anything about. If you apply $f$ to that, you'll get... a bunch of other things, $f(f(X))$, that you don't know anything about. $X$ is just $(X, 1)$, and $f(X)$ is just $(X, 2)$, etc. –  Qiaochu Yuan Sep 29 '12 at 23:07
Perfect! That is exactly what I wanted, the intuition. Proofs in category theory can of course be pretty opaque, but to understand it I like to have some intuition about the objects to the extent that it's possible. –  Adam Ray Sep 30 '12 at 0:26
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Let's be precise here. The category $\text{Dyn}$ of dynamical systems is the category whose objects are pairs $(X, f)$ where $X$ is a set and $f : X \to X$ is an endomorphism, and whose morphisms are maps of underlying sets $\phi : X_1 \to X_2$ such that $f_2 \circ \phi = \phi \circ f_1$. $\text{Dyn}$ has a forgetful functor $U : \text{Dyn} \to \text{Set}$ sending $(X, f)$ to $X$. The claim is that $U$ has both a left and a right adjoint.

The left adjoint sends a set $X$ to the dynamical system $(X \times \mathbb{N}, f)$ where $f(x, n) = (x, n+1)$ (exercise). This is what Lawvere and Schanuel mean by the "free dynamical system" on $X$.

The right adjoint sends a set $X$ to the dynamical system $(X^{\mathbb{N}}, f)$ of sequences $x(n)$ of elements of $X$, where $f(x(n)) = x(n+1)$ is the left shift (exercise). This is the "cofree dynamical system" on $X$.

You shouldn't be thinking of dynamical systems as graphs for the purposes of doing either of these exercises. There is a functor embedding $\text{Dyn}$ into the category of digraphs, but you have no guarantee that this embedding respects adjoints in any sense. Think, for example, of the embedding of abelian groups into groups.

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On pp.316-317 Lawvere explains what he means by a chaotic map of sets: Let < X ,/alpha> be a dynamical system, and let f be a map from X to a set Y . (Think of Y as being a set of values for some measurement of X , he suggests.) Now f is chaotic if for every possible sequence of values in Y ^ N , there is at least one state x of X whose "future" /alpha^ n x) maps to it. –  Adam Ray Sep 30 '12 at 0:51
Then on p.375 Lawvere says that the right adjoint to the forgetful functor from Dyn "produces the chaotic dynamical systems," specifically referring the reader back to the previous discussion, in which it was the map between sets--the "observation instrument" or "observable" as Lawvere calls it--that was called "chaotic." Can you explain this, as well as explaining the reason why such systems would be cofree? –  Adam Ray Sep 30 '12 at 0:54
Sorry about the coding on that first comment. You can probably tell what I meant at the end there, etc. –  Adam Ray Sep 30 '12 at 1:01
@Adam: aha. I am mistaken about the correct right adjoint. One moment. –  Qiaochu Yuan Sep 30 '12 at 1:02
@Adam: the definition of chaotic is essentially a restatement of the universal property of the right adjoint. I think this better worked out as an exercise than explained by someone else. Cofreeness here is morally a special case of the tensor-hom adjunction (en.wikipedia.org/wiki/Tensor-hom_adjunction). –  Qiaochu Yuan Sep 30 '12 at 1:08
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