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In this paper (Kock 2012), we see a data structure with circular symmetry. It is the cyclic list monad of Examples 3.10. The author is showing that data structures with symmetries can be cast as polynomial monads on the 2-category of groupoids. I would like to know the Eilenberg-Moore category for the cyclic list monad. I have a theory that the data structure for the quantum convex spaces, like finite dimensional spaces, is a data structure with a cyclic symmetry. I am thinking this because basic quantum experiments such as stern Gerlach or just polarization apparatuses have a circular symmetry. They have dials that rotate the basis of the measuring device.

Someone has suggested that Kock was writing about cyclic lists as a data structure defined by just an endofunctor, rather than a monad. I would be perfectly happy if anyone wants to give the EM category for that endofunctor, or the category of algebras for that endofunctor.

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    $\begingroup$ I don't see any mention of monad structure (and in any case I wouldn't know how that would go), but it is a polynomial endofunctor on groupoids. So maybe you want to know what is the category of algebras in the endofunctor sense? $\endgroup$
    – Todd Trimble
    Commented Aug 18, 2019 at 19:45
  • $\begingroup$ @ToddTrimble, Hi, I'm afraid I am not sure how just an endorfunctor can be a data structure, which is how I am seeing the lists. How can there be a category of algebras just for an endofunctor? Do you mean this as in contrast to an endo-2-functor? $\endgroup$
    – Ben Sprott
    Commented Aug 18, 2019 at 23:38
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    $\begingroup$ The notion of algebra of an endofunctor is very well-known in category theory, especially those aspects of category theory that touch on data types. Here a $T$-algebra for an endofunctor $T: C \to C$ is just an object $c$ of $C$ together with a map $Tc \to c$, with no other assumptions. Especially significant is whether initial $T$-algebras can be constructed, and many data types arise in just this way. For example, the initial algebra of the endofunctor defined by $T(c) = 1 + c \times c$ is the data type of binary trees. An old result of Lambek is: for $c$ initial, the structure map (cont.) $\endgroup$
    – Todd Trimble
    Commented Aug 18, 2019 at 23:51
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    $\begingroup$ $Tc \to c$ is necessarily an isomorphism, so $Tc \cong c$, or $c$ is a "fixed point" of the endofunctor $T$. For example, for the binary trees example, the structure map $Tc = 1 + c \times c \to c$ assigns to the element of $1$ the trivial tree without children, and to a pair of trees $(t, t')$ the tree whose root has left-child = root of $t$, and right-child = root of $t'$. (Notice that in this example, there is no monad structure at hand -- although I'd have to think to show that none exists.) The inverse $c \to Tc = 1 + c \times c$ takes a tree and deconstructs it into its left, right pair. $\endgroup$
    – Todd Trimble
    Commented Aug 18, 2019 at 23:55
  • $\begingroup$ @ToddTrimble, that's excellent. Thank you for sharing. I guesa I didn't read the paper closely enough to see that he may be talking about this construction. I am perfectly happy if someone gives the EM category for endofunctor of cyclic lists. $\endgroup$
    – Ben Sprott
    Commented Aug 19, 2019 at 1:52

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