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Michael Albanese
Michael Albanese
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There's certainly a homotopy-theoretic analogue. AA universal cover of a connected space X$X$ is (up to homotopy) a simply connected space X'$X'$ and a map X' -> X$X' \to X$ which is an isomorphism on πn$\pi_n$ for n >= 2$n \geq 2$. WeWe could next ask for a 2$2$-connected cover X''$X''$ of X'$X'$: a space X''$X''$ with πkX'' = 0$\pi_kX = 0$ for k <= 2$k \leq 2$ and a map X'' -> X'$X'' \to X'$ which is an isomorphism on πn$\pi_n$ for n >= 3$n \geq 3$. TheThe homotopy fiber of such a map will have a single nonzero homotopy group, in dimension 1-$1$ -it it will be a K(π2X, 1)$K(\pi_2X, 1)$.  (For the universal cover the fiber was the discrete space π1X = K(π1X, 0)$\pi_1X = K(\pi_1X, 0)$.)

An example is the Hopf fibration K(Z, 1) = S1 -> S3 -> S2$K(\mathbb{Z}, 1) = S^1 \to S^3 \to S^2$.

Geometrically it's harder to see what's going on with the 2$2$-connected cover than with the universal cover, because fibrations with fiber of the form K(G, 1)$K(G, 1)$ are harder to describe than fibrations with discrete fibers (covering spaces).

There's certainly a homotopy-theoretic analogue. A universal cover of a connected space X is (up to homotopy) a simply connected space X' and a map X' -> X which is an isomorphism on πn for n >= 2. We could next ask for a 2-connected cover X'' of X': a space X'' with πkX'' = 0 for k <= 2 and a map X'' -> X' which is an isomorphism on πn for n >= 3. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension 1--it will be a K(π2X, 1).  (For the universal cover the fiber was the discrete space π1X = K(π1X, 0).)

An example is the Hopf fibration K(Z, 1) = S1 -> S3 -> S2.

Geometrically it's harder to see what's going on with the 2-connected cover than with the universal cover, because fibrations with fiber of the form K(G, 1) are harder to describe than fibrations with discrete fibers (covering spaces).

There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi_n$ for $n \geq 2$. We could next ask for a $2$-connected cover $X''$ of $X'$: a space $X''$ with $\pi_kX = 0$ for $k \leq 2$ and a map $X'' \to X'$ which is an isomorphism on $\pi_n$ for $n \geq 3$. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension $1$ - it will be a $K(\pi_2X, 1)$. (For the universal cover the fiber was the discrete space $\pi_1X = K(\pi_1X, 0)$.)

An example is the Hopf fibration $K(\mathbb{Z}, 1) = S^1 \to S^3 \to S^2$.

Geometrically it's harder to see what's going on with the $2$-connected cover than with the universal cover, because fibrations with fiber of the form $K(G, 1)$ are harder to describe than fibrations with discrete fibers (covering spaces).

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Reid Barton
Reid Barton
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There's certainly a homotopy-theoretic analogue. A universal cover of a connected space X is (up to homotopy) a simply connected space X' and a map X' -> X which is an isomorphism on πn for n >= 2. We could next ask for a 2-connected cover X'' of X': a space X'' with πkX'' = 0 for k <= 2 and a map X'' -> X' which is an isomorphism on πn for n >= 3. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension 1--it will be a K(π2X, 1). (For the universal cover the fiber was the discrete space π1X = K(π1X, 0).)

An example is the Hopf fibration K(Z, 1) = S1 -> S3 -> S2.

Geometrically it's harder to see what's going on with the 2-connected cover than with the universal cover, because fibrations with fiber of the form K(G, 1) are harder to describe than fibrations with discrete fibers (covering spaces).