of ${\operatorname}{Span}_k({\mathcal}{X},X_{\bullet,\dots,\bullet})$ to $X_{\bullet,\dots,\bullet}\in{CSS}_k({\mathcal}{X})$ depends continuously on ${\mathcal}{X}$ and $X_{\bullet,\dots,\bullet}$ (i.e. it preserves small limits). This result is somewhat more subtle than it first appears: for example, the functor $${CSS}_k({\mathcal}{X})\xrightarrow{X_{\bullet,\dots,\bullet}\mapsto {\operatorname}{Span}_k({\mathcal}{X},X_{\bullet,\dots,\bullet})}{CSS}_k(\hat{\mathscr{S}})$$ is not continuous - it preserves neither products nor the terminal object. To correctly understand the continuity of ${\operatorname}{Span}_k$, we need to work in a larger context: we assemble all the $\infty$-categories ${CSS}_k({\mathcal}{X})$ into one large $\infty$-category $\int{CSS}_k$, whose (roughly speaking)[^1]

-   objects are pairs $({\mathcal}{X},X_{\bullet,\dots,\bullet})$, where ${\mathcal}{X}$ is an arbitrary $\infty$-topos and $X_{\bullet,\dots,\bullet}\in {CSS}_k({\mathcal}{X})$ is a higher category internal to ${\mathcal}{X}$, and

-   morphisms $({\mathcal}{X},X_{\bullet,\dots,\bullet})\to ({\mathcal}{Y},Y_{\bullet,\dots,\bullet})$ consist of a geometric morphism of $\infty$-topoi, $(f^*\dashv f_*):{\mathcal}{X}\leftrightarrows{\mathcal}{Y}$, together with a morphism $X_{\bullet,\dots,\bullet}\to (f_*)_!Y_{\bullet,\dots,\bullet}$ in ${CSS}_k({\mathcal}{X})$.

In Theorem \[thm:SpanCont\], we prove that the functor $$\int{CSS}_k\xrightarrow{({\mathcal}{X},X_{\bullet,\dots,\bullet})\mapsto {\operatorname}{Span}_k({\mathcal}{X},X_{\bullet,\dots,\bullet})}{CSS}_k(\hat{\mathscr{S}})$$ is continuous (preserves small limits).

Consequently, suppose that for every $U\in {\mathcal}{X}$ we assign (in a natural way) a complete Segal object $\sigma(U)\in {CSS}_k({\mathcal}{X}_{/U})$, in a manner which depends locally on $U\in{\mathcal}{X}$: that is, for any colimit diagram $U_i\to U$ in ${\mathcal}{X}$ indexed by a small simplicial set $i\in I$, $$\sigma(U)=\lim_{i\in I}\sigma(U_i)$$ (where the latter limit is taken in $\int{CSS}_k$), then

\[eq:IntroSpanStack\] $${\mathcal}{X}^{op}\xrightarrow{U\mapsto {\operatorname}{Span}_k({\mathcal}{X}_{/U},\sigma(U))}{CSS}_k(\hat{\mathscr{S}})$$ defines an $(\infty,k)$-stack over ${\mathcal}{X}$ (cf. Theorem \[thm:StackofSpans\]). As a first 