Proposition 3.30.
Given a morphism \(f\colon X \to Y\), there is a long exact sequence of pointed objects in \((T_{/X})_{\leq 0}\) of the form
\[\dots \to \pi_n(f) \to \pi_n(X) \to f^*\pi_n(Y) \to \pi_{n-1}(f) \to \dots .\]
Here, where three consecutive terms are group objects, exactness has its usual internal meaning: the image of one morphism is the kernel of the next. At degree zero, the group object \(f^*\pi_1(Y)\) acts on the pointed object \(\pi_0(f)\), and the fiber of \(\pi_0(f)\to\pi_0(X)\) over the basepoint is the orbit of the basepoint under this action.
Proof
When \(T = \An\), this is the usual long exact sequence of a homotopy fiber, including its group action on \(\pi_0\) in the final nonabelian degree. The claim follows pointwise for any presheaf topos \(\PSh(C)=\Fun(C\catop,\An)\). By Theorem 2.42, an arbitrary topos is a left exact localization of a presheaf topos. The localization preserves the finite limits, colimits, and homotopy group objects used in the construction, by Lemma 3.25, so it carries the pointwise long exact sequence to the asserted sequence in \(T\). This is the argument of [Lurie 2009, Remark 6.5.1.5].
References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.