ScalingStacks

Definition 14.1 ([4]). Let $\CSS(\Delta^{0})$ be the $\infty$-category $\mathcal{S}$ of Kan simplicial sets. Suppose now that $n$ is a positive integer; assume that both a presentable $\infty$-category $\CSS(\Delta^{\!\times n-1})$ and a fully faithful functor

$$c_{n-1}\colon\CSS(\Delta^{0})\hookrightarrow\CSS(\Delta^{\!\times n-1})$$

that preserves all small colimits have been constructed. Let us call a simplicial object $X\colon\mathrm{N}\Delta^{\mathrm{op}}\to\CSS(\Delta^{\!\times n-1})$ an $n$-fold Segal space if it satisfies the following pair of conditions.

1.   (B.1)

   The object $X_{0}$ lies in the essential image of $c_{n-1}$.

2.   (B.2)

   For any integers $0<k<m$, the object $X_{m}$ is exhibited as the limit of the diagram

   $$X(\{0,1,\dots,k\})\rightarrow X(\{k\})\leftarrow X(\{k,k+1,\dots,m\}).$$

Now for any $n$-fold Segal space $X$, one may apply the right adjoint to the functor $c_{n-1}$ objectwise to $X$ to obtain a simplicial space $\iota_{1}X$. Let us call $X$ an $n$-fold complete Segal space if it satisfies the following additional condition.

1.   (B.3)

   The Kan complex $(\iota_{1}X)_{0}$ is exhibited as the limit of the composite functor

   $$\Delta_{/\mathrm{N}E}^{\mathrm{op}}\to\Delta^{\mathrm{op}}\stackrel{{\scriptstyle\iota_{1}X}}{{\longrightarrow}}\CSS_{0},$$

where the category $E$ is as in Ex. 2.5. Denote by $\CSS(\Delta^{\!\times n})$ the full subcategory of $\Fun(\mathrm{N}\Delta^{\mathrm{op}},\CSS(\Delta^{\!\times n-1})$ spanned by the $n$-fold complete Segal spaces.