Definition 5.88.
Let \(T\) be a topos. We define the \(n\)-th Goodwillie approximation of \(T\) as
\[T^{(n)} := T/(\Conn_{\infty})^{n+1}.\]
This gives rise to a tower of logoi
\[T \to \dots \to T^{(2)} \to T^{(1)} \to T^{(0)} \to T^{(-1)} = *.\]
By passing to right adjoints (geometric morphisms), this may alternatively be regarded as a filtration of \(T\) by subtopoi:
\[* = T^{(-1)} \hookrightarrow T^{(0)} \hookrightarrow T^{(1)} \hookrightarrow \dots \hookrightarrow T.\]