One of the axioms for $(\infty,1)$-topoi is that the topos is disjoint, meaning that we have the following pullback diagram
$$
\begin{matrix}
0 & \rightarrow & A \\
\downarrow & & \downarrow \\
B & \rightarrow & A \coprod B
\end{matrix}
$$
What is a non-example of an $(\infty,1)$-topos where this fails?



What is a non-example of and $(\infty,1)$-topos where disjointness fails?
A basic non-example is the $(\infty,1)$-category of based spaces. Take $A = B = S^1$, then the homotopy pullback of the two inlcusions $S^1 \to S^1 \vee S^1$ is disconnected. In fact, it consists of countably many contractible components.


