Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the

Segal condition: For each $j$, the map $$ X_j \to \underbrace{X_1 \times_{X_0} X_1 \times_{X_0} \dots \times_{X_0} X_1}_j$$ reading off the $j$ main edges of a $j$-simplex (i.e. the ones connecting vertex $\ell$ to vertex $\ell+1$) is a homotopy equivalence of spaces.

Here and throughout this question, my convention is that all fibered products are homotopy fibered products.

Any Segal space $X$ determines a homotopy category $\mathrm h_1 X$ whose object set is the underlying set of $X_0$ and the morphism set from $x\in X_0$ to $y\in X_0$ is $$ \hom_{\mathrm h_1 X}(x,y) = \pi_0 \left( \{x\} \times_{X_0} X_1 \times_{X_0} \{y\} \right) $$ (the two maps $X_1 \to X_0$ are the two face maps), and composition is given by applying $\pi_0$ (which note takes homotopy equivalences to isomorphisms of sets) to the zig-zag $$ \begin{align*} & \{x\} \times_{X_0} X_1 \times_{X_0} \{y\} \times \{y\} \times_{X_0} X_1 \times_{X_0} \{z\} \\ &\hspace{2in} \to \{x\} \times_{X_0} X_1 \times_{X_0} X_1 \times_{X_0} \{z\} \\ &\hspace{2in} \overset\sim\leftarrow \{x\} \times_{X_0} X_2 \times_{X_0} \{z\} \\ &\hspace{2in} \to \{x\} \times_{X_0} X_1 \times_{X_0} \{z\} \end{align*} $$ where the last arrow is the third face map.

A Segal space is complete if it satisfies

Completeness: Denote by $(X_1)^{\mathrm{inv}}$ the subspace of $X_1$ consisting of those points whose image in (the morphism set of) $\mathrm{h}_1X$ is invertible. The degeneracy map $$ X_0 \to (X_1)^{\mathrm{inv}}$$ is a homotopy equivalence of spaces.

Generalizing these notions, an $n$-uple Segal space is an $n$-times simplicial space, $X : \Delta^{\times n} \to \mathrm{Spaces}$, $\vec\bullet \mapsto X_{\vec \bullet}$, $\vec\bullet \in \mathbb N^{\times n}$, such that for each $i_1,\dots,i_{m-1},i_{m+1},\dots,i_n$, the simplicial space $X_{i_1,\dots,i_{m-1},\bullet,i_{m+1},\dots,i_n}$ is Segal. It is complete if the Segal space $X_{i_1,\dots,i_{m-1},\bullet,i_{m+1},\dots,i_n}$ is complete for each $i_1,\dots,i_{m-1},i_{m+1},\dots,i_n$.

An $n$-uple Segal space is $n$-fold if it additionally satisfies the condition of

Essential constancy: For each $i_1,\dots,i_{m-1},i_{m+1},\dots,i_n$, the degeneracy map $$ X_{i_1,\dots,i_{m-1},0,0,\dots,0} \to X_{i_1,\dots,i_{m-1},0,i_{m+1},\dots,i_n} $$ is a homotopy equivalence of spaces.

The name "uple" comes from a recent paper by Haugseng. When an $n$-uple Segal space is $n$-fold, it is called an $n$-fold Segal space. $n$-fold Segal spaces were invented to make sense of $(\infty,n)$-categories (the complete ones actually model $(\infty,n)$-categories, as do some other variations); the space of $m$-morphisms is $X_{1,\dots,1,0,\dots,0}$, where there are $m$ $1$s.

My question is the following:

Suppose I have an $n$-fold Segal space, i.e. I know it is an $n$-uple simplicial space, it is Segal in each index holding all other indices fixed, and satisfies essential constancy. What is a minimal collection of maps I need to check to know that it is complete?

For example, I want to expect that Segality in the other variables means that I only need to check that $X_{i_1,\dots,i_{m-1},\bullet,i_{m+1},\dots,i_n}$ is complete when all $i$s are $0$s or $1$s. Actually, I want to expect that that's true for $n$-uple Segal spaces, not just $n$-fold ones. This should require that "invertibility up to homotopy" is preserved by fiber products.

Then I want essential constancy to imply that actually it's enough to check, for each $m$, the case when $i_1 = \dots = i_{m-1} = 1$ and $i_{m+1} = \dots = i_n = 0$. This, if true, is a bit more subtle, since it uses compositions in one direction to say something about compositions in another direction. So it's a sort of Eckmann–Hilton argument.

Do my expectations match reality?


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