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Stream: theory: algebraic topology

Topic: Relating language shuffle to cartesian product

Naso (Dec 09 2021 at 15:17):

Let $M$ be a monoid, $NM$ its simplicial nerve and $A = \mathsf{Sub}(NM)$ its heyting algebra of subobjects.

For $X, Y \in A$ we can define an $\mathbb{N}$-indexed set $(X * Y)_n = \bigcup \{ x * y \mid x \in X_i, y \in Y_j, i + j = n \}$ where $x * y \subseteq M^n$ is the set of shuffles of $x$ and $y$.

We can then define $X \mid Y := \mathsf{cl}(X * Y)$ where $\mathsf{cl}(Z) \in A$ is the smallest simplicial subset such that $Z_n \subseteq \mathsf{cl}(Z)_n$ in $A$ (well-defined since $A$ is a complete lattice).

On the other hand, the maximal nondegenerate dimensional $n$-cells of the cartesian product $X \times Y$ can be identified with the set of shuffles $\bigcup \{ x * y \mid x \in X_i, y \in Y_j, i + j = n \}$

Vague question: can $X \mid Y$ be described constructively in terms of $X \times Y$ rather than by the closure operation $\mathsf{cl}$?

Is it the image of the composite of the left-hand maps in below? (Where $x, y$ are the inclusion maps,
$q : M \vee M \to M \times M$ is the quotient map from the free product of $M$ with itself to the cartesian product, and $\pi : M \vee M \to M$ is the projection.) image.png

Morgan Rogers (he/him) (Dec 09 2021 at 15:27):

Nasos Evangelou-Oost said:

Let $M$ be a monoid, $NM$ its simplicial nerve and $A = \mathsf{Sub}(NM)$ its heyting algebra of subobjects.

Subobjects in the category of simplicial sets, presumably?

For $X, Y \in A$ we can define an $\mathbb{N}$-indexed set $(X * Y)_n = \bigcup \{ x * y \mid x \in X_i, y \in Y_j, i + j = n \}$ where $x * y \subseteq M^n$ is the set of shuffles of $x$ and $y$.

What's a shuffle?

We can then define $X \mid Y := \mathsf{cl}(X * Y)$ where $\mathsf{cl}(Z) \in A$ is the smallest simplicial subset such that $Z_n \subseteq \mathsf{cl}(Z)_n$ in $A$ (well-defined since $A$ is a complete lattice).

If I understand this correctly, you're saying $(X*Y)_n$ is a subset of $(NM)_n$ for each $n$, and you're asking for the simplicial subset of $NM$ generated by the elements of these subsets?

Naso (Dec 09 2021 at 15:32):

Morgan Rogers (he/him) said:
"Subobjects in the category of simplicial sets, presumably?"

Yes

"What's a shuffle?"

I mean a shuffle product kind of as in https://en.wikipedia.org/wiki/Shuffle_algebra#Shuffle_product , but with a union of words rather than a formal sum. Sorry I'm not sure how to define it concisely, but do you see what I mean?

"If I understand this correctly, you're saying $(X*Y)_n$ is a subset of $(NM)_n$ for each $n$, and you're asking for the simplicial subset of $NM$ generated by the elements of these subsets?"

Yes