In algebraic topology, we have the map lifting lemma which says that given a covering space $p:(\tilde{X},\tilde{x})\rightarrow (X,x)$ and a map $f:(Y,y)\rightarrow (X,x)$ with $Y$ connected and locally path-connected, a lift $\tilde{f}:(Y,y)\rightarrow(\tilde{X},\tilde{x})$ of $f$ exists if and only if $f\_\*(\pi\_1(Y,y))\subset p\_\*(\pi\_1(\tilde{X},\tilde{x}))$.


I want to ask whether analogues of the above are true for finite etale maps and etale fundamental groups.



Map Lifting lemma and Etale fundamental group
The answer is yes, at least for finite covers (or even pro-finite covers). The key is to carefully translate everything into the language of finite $\pi\_1$-sets using the Galois correspondence. The canonical reference is of course SGA 1, Expose V - in particular $\S5,6$ and 7 once you're familiar with the basics.


By the universal property of fiber products, the existence of such a lifting $\tilde{f}$ is equivalent to the existence of a section of the covering map (ie finite etale morphism) $p\_Y : Y\times\_X\tilde{X}\rightarrow Y$ which contains the geometric point $(y,\tilde{x})$ (such a section if it exists, is unique).


By Galois theory, the pointed covering $p : (\tilde{X},\tilde{x})\rightarrow (X,x)$ is equivalent to the data of the pointed set $(p^{-1}(x),\tilde{x})$, together with the monodromy action of $\pi\_1(X,x)$ on the underlying set $p^{-1}(x)$. The pullback $p\_Y$ is (via the Galois correspondence) given by the pointed set $(p\_Y^{-1}(y),(y,\tilde{x}))$ together with the monodromy action of $\pi\_1(Y,y)$ acting on the underlying set, which is described by the map
$$\pi\_1(Y,y)\stackrel{f\_\*}{\rightarrow}\pi\_1(X,x)\rightarrow Aut(p^{-1}(x))\cong Aut(p\_Y^{-1}(y))\qquad (\*)$$
where the last isomorphism is induced by the natural bijection $p\_Y^{-1}(y)\cong p^{-1}(x)$.


(In short, the pullback of a cover over $X$ given by a $\pi\_1(X)$-set $T$ by any morphism $Y\rightarrow X$ is the cover over $Y$ given by pulling back the $\pi$-set structure on $T$ via the morphism $\pi\_1(Y)\rightarrow\pi\_1(X)$. (What else could it be?) See $\S6$ in SGA 1 Expose V)


Under these identifications, $\pi\_1(\tilde{X},\tilde{x})$ is naturally identified with the subgroup of $\pi\_1(X,x)$ defined by $Stab\_{\pi\_1(X,x)}(\tilde{x})$. The map $p\_Y$ admits a section containing the geometric point $(y,\tilde{x})$ if and only if the point $(y,\tilde{x})\in p\_Y^{-1}(y)$ is fixed by the monodromy action of $\pi\_1(Y,y)$ (Note that a section of $p\_Y$ is a morphism of finite etale covers over $Y$, and hence corresponds via the Galois correspondence to a $\pi\_1(Y,y)$-set morphism from the singleton set with trivial $\pi$-action to $p\_Y^{-1}(y)$. Since this morphism is a morphism of $\pi\_1(Y,y)$-sets, and hence must commute with the $\pi\_1(Y,y)$-action, it implies that the image of this map is fixed by $\pi\_1(Y,y)$).


However, by the definition of this monodromy action in $(\*)$, we see that this is true if and only if $f\_\*\pi\_1(Y,y)\subset Stab\_{\pi\_1(X,x)}(\tilde{x}) = \pi\_1(\tilde{X},\tilde{x})$.


