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3.5 Truncations and Postnikov Towers

Let $(X,x)$ be a pointed Kan complex. In §3.2.2, we introduced a sequence of groups

\[ \{ \pi _{m}(X,x) \} _{m > 0} \]

called the homotopy groups of $(X,x)$. These groups are very useful tools for analyzing the homotopy type of $X$. For example, the Kan complex $X$ is contractible if and only if it connected and all of its homotopy groups are trivial (Theorem 3.2.4.3). This is a special case of the following more general result, which classifies Kan complexes having (at most) one nontrivial homotopy group:

Proposition 3.5.0.1. Let $X$ be a connected Kan complex, let $x \in X$ be a vertex, and let $n$ be a positive integer. Suppose that the homotopy groups $\pi _{m}(X,x)$ vanish for every positive integer $m \neq n$. Then $X$ is homotopy equivalent to an Eilenberg-MacLane space $\mathrm{K}(G,n)$ for some group $G$ (which is abelian if $n \geq 2$).

We will prove Proposition 3.5.0.1 in §3.5.7 (see Corollary 3.5.7.18). To carry out the proof, it will be useful to break the hypothesis of Proposition 3.5.0.1 into two parts. In what follows, we fix a positive integer $n$.

  • We say that a Kan complex $X$ is $n$-connective if it is connected and the homotopy group $\pi _{m}(X,x)$ vanishes for every integer $0 < m < n$ and every choice of base point $x \in X$.

  • We say that a Kan complex $X$ is $n$-truncated if the homotopy group $\pi _{m}(X,x)$ vanishes for every integer $m > n$ and every choice of base point $x \in X$.

Stated more informally, a Kan complex $X$ is $n$-connective if its homotopy groups are concentrated in degrees $\geq n$, and $n$-truncated if its homotopy groups are concentrated in degrees $\leq n$. Each of these conditions admits a number of equivalent formulations, which we study in §3.5.1 and §3.5.7, respectively.

Proposition 3.5.0.1 asserts that if a Kan complex $X$ is both $n$-connective and $n$-truncated, then it is homotopy equivalent to an Eilenberg-MacLane space $\mathrm{K}(G,n)$. We will deduce this from a structural analysis of $n$-truncated Kan complexes in general. We begin by observing that $X$ is $n$-truncated if and only if, for every integer $m \geq n+2$, the restriction map

\[ \theta _{m}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, X) \]

is surjective (see Proposition 3.5.7.7). In §3.5.3, §3.5.4, and §3.5.5, we study stronger versions of this condition:

  • We say that $X$ is $(n+1)$-coskeletal if, for every integer $m \geq n+2$, the map $\theta _ m$ is a bijection (Definition 3.5.3.1).

  • We say that $X$ is weakly $n$-coskeletal if it is $(n+1)$-coskeletal and, in addition, the map $\theta _{n+1}$ is injective (Definition 3.5.4.1).

  • We say that $X$ is an $n$-groupoid if it is weakly $n$-coskeletal and every $n$-simplex $\sigma : \Delta ^ n \rightarrow X$ is determined by its homotopy class relative to $\operatorname{\partial \Delta }^{n}$ (see Definition 3.5.5.1 and Proposition 3.5.5.12).

For any Kan complex $X$, we have the following implications:

\[ \xymatrix@R =50pt@C=50pt{ \textnormal{$X$ is an $n$-groupoid} \ar@ {=>}[d] \\ \textnormal{$X$ is weakly $n$-coskeletal} \ar@ {=>}[d] \\ \textnormal{$X$ is $(n+1)$-coskeletal} \ar@ {=>}[d] \\ \textnormal{$X$ is $n$-truncated}. } \]

None of these implications is reversible. However, they are reversible “up to homotopy” in the following sense: every $n$-truncated Kan complex is homotopy equivalent to an $n$-groupoid. More generally, in §3.5.6 we will associate to any Kan complex $X$ an $n$-groupoid $\pi _{\leq n}(X)$, which we refer to as the fundamental $n$-groupoid of $X$ (Construction 3.5.6.10). It is equipped with a comparison map $f: X \rightarrow \pi _{\leq n}(X)$ which is universal among maps from $X$ to $n$-groupoids (Proposition 3.5.6.5), which is a homotopy equivalence if and only if $X$ is $n$-truncated (Variant 3.5.7.16).

Remark 3.5.0.2. The preceding characterization of $n$-truncated Kan complexes has a counterpart for $n$-connective Kan complexes. A Kan complex $X$ is $n$-connective if and only if it is homotopy equivalent to a Kan complex $Y$ having a single $m$-simplex for each $m < n$ (Proposition 3.5.2.9 and Remark 3.5.2.10). We will prove Proposition 3.5.0.1 by showing that, in this case, the fundamental $n$-groupoid $\pi _{\leq n}(Y)$ is isomorphic to an Eilenberg-MacLane space $\mathrm{K}(G,n)$, for some group $G$. See Proposition 3.5.5.16.

For any Kan complex $X$, the collection of fundamental $n$-groupoids $\{ \pi _{\leq n}(X) \} $ can be organized into an inverse system

\[ \cdots \rightarrow \pi _{\leq 3}(X) \rightarrow \pi _{\leq 2}(X) \rightarrow \pi _{\leq 1}(X) \rightarrow \pi _{0}(X) \]

which we will refer to as the (canonical) Postnikov tower of $X$ (Example 3.5.8.2). In §3.5.8, we show that each of the transition maps $\pi _{\leq n}(X) \rightarrow \pi _{\leq n-1}(X)$ is a Kan fibration, whose fiber over a vertex $x$ is homotopy equivalent to the Eilenberg-MacLane space $\mathrm{K}(G,n)$ for $G = \pi _{n}(X,x)$ (Corollary 3.5.8.9). Stated more informally, every Kan complex $X$ can be built as a successive extension of Eilenbeg-MacLane spaces.

For many applications, it will be useful to work with relative versions of the preceding conditions. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Let us assume for simplicity that $f$ is a Kan fibration, so that the fiber $X_{y} = \{ y\} \times _{Y} X$ is a Kan complex for each vertex $y \in Y$. We will say that $f$ is $n$-connective if each of the Kan complexes $X_{y}$ is $n$-connective (see Definition 3.5.1.13 and Proposition 3.5.1.22), and we say that $f$ is $n$-truncated if each of the Kan complexes $X_{y}$ is $n$-truncated (see Definition 3.5.9.1 and Proposition 3.5.9.8). In §3.5.2 and §3.5.9, we study a number of different formulations of these conditions. In particular, we show that both are characterized by lifting properties:

  • A Kan fibration $f: X \rightarrow Y$ is $n$-connective if and only if the every lifting problem

    3.64
    \begin{equation} \begin{gathered}\label{equation:connective-as-lifting} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & X \ar [d]^{f} \\ B \ar [r] \ar@ {-->}[ur] & Y } \end{gathered} \end{equation}

    admits a solution, provided that $B$ is a simplicial set of dimension $\leq n$ and $A$ is a simplicial subset of $B$. (Proposition 3.5.2.1).

  • A Kan fibration $f: X \rightarrow Y$ is $n$-truncated if and only if every lifting problem (3.64) has solution, provided that $A$ is a simplicial subset of $B$ which contains the $(n+1)$-skeleton of $B$ (Corollary 3.5.9.23).

Structure

  • Subsection 3.5.1: Connectivity
  • Subsection 3.5.2: Connectivity as a Lifting Property
  • Subsection 3.5.3: Coskeletal Simplicial Sets
  • Subsection 3.5.4: Weakly Coskeletal Simplicial Sets
  • Subsection 3.5.5: Higher Groupoids
  • Subsection 3.5.6: Higher Fundamental Groupoids
  • Subsection 3.5.7: Truncated Kan Complexes
  • Subsection 3.5.8: The Postnikov Tower of a Kan Complex
  • Subsection 3.5.9: Truncated Morphisms