The Stacks project

Theorem 57.6.2. Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$.

  1. The fibre functor $F_{\overline{x}}$ defines an equivalence of categories

    \[ \textit{FÉt}_ X \longrightarrow \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} \]
  2. Given a second geometric point $\overline{x}'$ of $X$ there exists an isomorphism $t : F_{\overline{x}} \to F_{\overline{x}'}$. This gives an isomorphism $\pi _1(X, \overline{x}) \to \pi _1(X, \overline{x}')$ compatible with the equivalences in (1). This isomorphism is independent of $t$ up to inner conjugation.

  3. Given a morphism $f : X \to Y$ of connected schemes denote $\overline{y} = f \circ \overline{x}$. There is a canonical continuous homomorphism

    \[ f_* : \pi _1(X, \overline{x}) \to \pi _1(Y, \overline{y}) \]

    such that the diagram

    \[ \xymatrix{ \textit{FÉt}_ Y \ar[r]_{\text{base change}} \ar[d]_{F_{\overline{y}}} & \textit{FÉt}_ X \ar[d]^{F_{\overline{x}}} \\ \textit{Finite-}\pi _1(Y, \overline{y})\textit{-Sets} \ar[r]^{f_*} & \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} } \]

    is commutative.

Proof. Part (1) follows from Lemma 57.5.5 and Proposition 57.3.10. Part (2) is a special case of Lemma 57.3.11. For part (3) observe that the diagram

\[ \xymatrix{ \textit{FÉt}_ Y \ar[r] \ar[d]_{F_{\overline{y}}} & \textit{FÉt}_ X \ar[d]^{F_{\overline{x}}} \\ \textit{Sets} \ar@{=}[r] & \textit{Sets} } \]

is commutative (actually commutative, not just $2$-commutative) because $\overline{y} = f \circ \overline{x}$. Hence we can apply Lemma 57.3.11 with the implied transformation of functors to get (3). $\square$

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