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I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where $X$ is a nice algebraic variety over complex numbers $\mathbb{C}$.

Is it reasonable to hope for a definition of higher algebraic homotopy groups $\pi_{n}^{alg}(X)$ of a scheme $X$ ?

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    $\begingroup$ There are higher algebraic homotopy groups of schemes. This was first developed by Artin-Mazur, and then developed further by Eric Friedlander. Recently, Michael McQuillan has proposed a different definition of the second homotopy group. $\endgroup$ – Jason Starr Jan 20 '16 at 14:56
  • $\begingroup$ @JasonStarr does it recover the relation between $\pi_{1}$ and $\pi_{1}^{alg}$ ? Could you put the references ? $\endgroup$ – Ofra Jan 20 '16 at 15:02
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    $\begingroup$ en.wikipedia.org/wiki/%C3%89tale_homotopy_type $\endgroup$ – Qiaochu Yuan Jan 20 '16 at 15:45
  • $\begingroup$ ncatlab.org/nlab/show/étale+homotopy $\endgroup$ – Urs Schreiber Jan 21 '16 at 14:05
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I am writing the comments above so that this question does not remain unanswered. As Qiaochu Yuan writes, one good resource is the wikipedia page for etale homotopy theory. Here are two references from MathSciNet.

MR0883959 (88a:14024)
Artin, M.; Mazur, B.
Etale homotopy.
Lecture Notes in Mathematics, 100. Springer-Verlag, Berlin, 1986. iv+169 pp.
ISBN: 3-540-04619-4

MR0676809 (84h:55012)
Friedlander, Eric M.
Étale homotopy of simplicial schemes.
Annals of Mathematics Studies, 104. Princeton University Press, Princeton, N.J.;
University of Tokyo Press, Tokyo, 1982. vii+190 pp.
ISBN: 0-691-08288-X; 0-691-08317-7

Here is the URL for the article of Michael McQuillan: Elementary Topology of Champs.

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