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I suspect that this has been addressed somewhere already, but I cannot find anything. Let $\hat{\Delta}$ denote simplicial sets, and $Mon\hat{\Delta}$ simplicial monoids. There is a forgetful functor $(-):Mon\hat{\Delta}\to\hat{\Delta}$ which admits a left adjoint $-^{*}:\hat{\Delta}\to Mon\hat{\Delta}$. My idea is to define a sort of "boundary" operator as a natural endotransformation of $-^{*}$, whose component at each $X:\hat{\Delta}$ is the function sending lists of n-cells to the concatenation of the corresponding list of lists of faces of each n-cell. It's easy to see that this a shifted homomorphism of simplicial monoids. I don't feel safe calling it a boundary since the free monoid discerns between $[x]$ and $[x,x]$ (these are living in some grade of $X^{*}$), and therefor the boundary of closed structures is captured by sequential redundancies rather than simply vanishing algebraically. I want to keep track of this redundancy rather than quotienting mindlessly, since part of my motivation is to recover the homotopical information which is lost when abelianizing into homology. Using noncommutative monoids maintains harmony with the simplex category that locally models simplicial sets, by allowing the ordering of simplicial faces (associated to cohomotopic orientation via the homotopy correspondence, see the Hopf degree theorem) to interact politely with the synthetic algebra of the concatenation of lists of cells. We exfoliate the structure condensed into a simplicial set $X$ by "dissolving" or analysing the ordered (simplicial) stuff in a bath of freely-orderable (listable) bits. The full picture of this will be lucid given the universal property of $\Delta_{\alpha}$ as the free "monoidally-pointed category" (monoidal category equipped with a monoid in it) on a point, so perhaps I would be better suited working with augmented simplicial sets to answer my question.

My question is whether this list-of-lists-of-faces-of-lists-of-cells operation is documented anywhere, and whether it has any notable properties to justify its study in the context of (co)homotopy?

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    $\begingroup$ How is the "list of lists of faces of each n-cell" defined? $\endgroup$ Feb 18, 2021 at 18:50
  • $\begingroup$ Every n-cell in $X$ has a list of faces of length n+1, which is an (n-1)-cell in the free simplicial monoid $X^{*}$. This defines a map $X\to \Sigma X^{*}$, which is the adjunct to a map $X^{*}\to \Sigma X^{*}$. This operation is defined on lists of n-cells, giving you the result of first replacing each list element with its list of faces, and then passing this list-of-lists through the component for the multiplication of the free monoid monad, ie just concatenate the finite list of finite lists (all of whose elements are faces of n-cells). $\endgroup$ Feb 19, 2021 at 2:09
  • $\begingroup$ Okay, how is a "shifted homomorphism of simplicial monoids" defined? $\endgroup$ Feb 19, 2021 at 2:20
  • $\begingroup$ It's just a map into the suspension. I am sorry if my terminology is incorrect. The n-cells of $X^{*}$ are lists of n-cells in $X$, and the suspension shifts the grading by one. $\endgroup$ Feb 19, 2021 at 10:20
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    $\begingroup$ There is more than one suspension and loop functor in simplicial homotopy theory, but none of them simply shifts simplicial degrees by one. Additionally, for simplicial monoids the suspension functor must be derived, if you are to get a correct answer. If you simply shift simplicial degrees by one, how is the resulting sequence of sets turned into a simplicial set? $\endgroup$ Feb 19, 2021 at 18:22

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