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3.5.6 Higher Fundamental Groupoids

Let $X$ be a Kan complex. Recall that the fundamental groupoid $\pi _{\leq 1}(X)$ is a category whose objects are the vertices of $X$, and whose morphisms are given by homotopy classes of paths (Definition 1.4.6.12). We now consider a higher-dimensional version of this construction.

Definition 3.5.6.1. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. We say that a morphism of Kan complexes $f: X \rightarrow Y$ exhibits $Y$ as a fundamental $n$-groupoid of $X$ if the following conditions are satisfied:

$(a)$

The simplicial set $Y$ is an $n$-groupoid (Definition 3.5.5.1).

$(b)$

The morphism $f$ is bijective on $m$-simplices for $m < n$.

$(c)$

The morphism $f$ is surjective on $n$-simplices.

$(d)$

If $\sigma $ and $\sigma '$ are $n$-simplices of $X$ satisfying $f(\sigma ) = f(\sigma ')$, then $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{\partial \Delta }^{n}$ (see Definition 3.2.1.3).

Remark 3.5.6.2. Let $n$ be a nonnegative integer and let $f: X \rightarrow Y$ be a morphism of Kan complexes which exhibits $Y$ as a fundamental $n$-groupoid of $X$. Then $f$ is a Kan fibration (Corollary 3.5.5.14). In particular, since it is surjective on vertices, it is surjective on $m$-simplices for every integer $m$ (see Remark 3.1.2.8).

Example 3.5.6.3. Let $X$ be a Kan complex and let $\pi _{\leq 1}(X)$ denote the fundamental groupoid of $X$ (Definition 1.4.6.12). Then the nerve $\operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ is a $1$-groupoid (Proposition 3.5.5.8). By construction, the tautological map $u: X \rightarrow \operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ is bijective on vertices and surjective on edges. Moreover, two edges of $X$ have the same image in $\operatorname{N}_{\bullet }( \pi _{\leq 1}(X) )$ if and only if they are homotopic relative to $\operatorname{\partial \Delta }^{1}$ (see Corollary 1.4.3.7). It follows that $u$ exhibits $\operatorname{N}_{\bullet }(\pi _{\leq 1}(X))$ as a fundamental $1$-groupoid of $X$.

Example 3.5.6.4. Let $n$ be a nonnegative integer and let $X$ be an $n$-groupoid. Then the identity map $\operatorname{id}_{X}: X \rightarrow X$ exhibits $X$ as a fundamental $n$-groupoid of itself.

Fundamental $n$-groupoids can be characterized by a universal mapping property.

Proposition 3.5.6.5. Let $n$ be a nonnegative integer and let $f: X \rightarrow Y$ be a morphism of Kan complexes which exhibits $Y$ as a fundamental $n$-groupoid of $X$. Then, for every $n$-groupoid $Z$, composition with $f$ induces an isomorphism of simplicial sets $\operatorname{Fun}(Y, Z) \rightarrow \operatorname{Fun}(X,Z)$.

Proof. Let $K$ be a simplicial set; we will show that precomposition with $f$ induces a bijection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Fun}(Y,Z) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K, \operatorname{Fun}(X,Z) )$. Replacing $Z$ by the $n$-groupoid $\operatorname{Fun}(K,Z)$ (Corollary 3.5.5.13), we are reduced to proving that the natural map

\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(Y, Z) \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(X,Z) \]

is a bijection.

Let $h: X \rightarrow Z$ be a morphism of Kan complexes; we wish to show that there is a unique morphism $g: Y \rightarrow Z$ satisfying $g \circ f = h$. We first claim that there is a unique morphism $g_0: \operatorname{sk}_{n}(Y) \rightarrow Z$ satisfying $g_0 \circ \operatorname{sk}_{n}(f) = h|_{ \operatorname{sk}_{n}(X) }$. Our assumption that $f$ exhibits $Y$ as a fundamental $n$-groupoid of $X$ guarantees that $\operatorname{sk}_{n}(f)$ is surjective. It will therefore suffice to show that if $\sigma $ and $\sigma '$ are $m$-simplices of $X$ for $m \leq n$ satisfying $f(\sigma ) = f(\sigma ')$, then $h(\sigma ) = h(\sigma ')$. If $m < n$, then $\sigma = \sigma '$ and the result is clear. In the case $m = n$, the assumption that $f(\sigma ) = f(\sigma ')$ guarantees that $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{\partial \Delta }^{n}$, in which case the desired result follows from Proposition 3.5.5.12.

It follows from Remark 3.5.6.2 that every $(n+1)$-simplex $\tau $ of $Y$ can be lifted to an $(n+1)$-simplex $\widetilde{\tau }$ of $X$. In particular, $g_0 \circ \tau |_{ \operatorname{\partial \Delta }^{n+1} }$ extends to an $(n+1)$-simplex of $Z$, given by $h( \widetilde{\tau } )$. Since $Z$ is weakly $n$-coskeletal (Proposition 3.5.5.10), Proposition 3.5.4.12 guarantees that $g_0$ extends uniquely to a morphism $g: Y \rightarrow Z$, which automatically satisfies the equation $g \circ f = h$. $\square$

Notation 3.5.6.6. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. We will see later that there exists a morphism of Kan complexes $f: X \rightarrow Y$ which exhibits $Y$ as a fundamental $n$-groupoid of $X$ (Theorem 3.5.6.17). It follows from Proposition 3.5.6.5 that $Y$ is unique up to (canonical) isomorphism and depends functorially on $X$. To emphasize this dependence, we will typically denote the simplicial set $Y$ by $\pi _{\leq n}(X)$, and refer to it as the fundamental $n$-groupoid of $X$.

Warning 3.5.6.7. Let $X$ be a Kan complex. We have now assigned two different meanings to the notation $\pi _{\leq 1}(X)$:

  • The fundamental groupoid of $X$ (Definition 1.4.6.12), which is a category.

  • The fundamental $1$-groupoid of $X$ (Notation 3.5.6.6), which is a simplicial set.

However, the danger of confusion is slight: by virtue of Example 3.5.6.3, the fundamental $1$-groupoid of $X$ is isomorphic to the nerve of the fundamental groupoid of $X$.

Example 3.5.6.8. Let $X$ be a Kan complex, let $\pi _0(X)$ be the set of connected components of $X$. Then the fundamental $0$-groupoid $\pi _{\leq 0}(X)$ can be identified with the constant simplicial set $\underline{ \pi _0(X) }$. More precisely, the tautological map $X \rightarrow \underline{ \pi _0(X) }$ exhibits $\underline{\pi _0(X) }$ as a fundamental $0$-groupoid of $X$ (see Proposition 3.5.5.7).

Example 3.5.6.9. Let $n \geq 0$ be an integer and let $X = \mathrm{K}( A_{\ast } )$ be the Eilenberg-MacLane space associated to a chain complex of abelian groups

\[ \cdots \rightarrow A_{n+1} \xrightarrow {\partial } A_{n} \xrightarrow {\partial } A_{n-1} \xrightarrow { \partial } A_{n-2} \rightarrow \cdots \]

(see Construction 2.5.6.3). Let $B_{n} \subseteq A_{n}$ denote the image of the differential $\partial : A_{n+1} \rightarrow A_{n}$, and let $A'_{\ast }$ denote the chain complex

\[ \cdots \rightarrow 0 \rightarrow A_{n} / B_{n} \rightarrow A_{n-1} \xrightarrow { \partial } A_{n-2} \rightarrow \cdots \]

Since $A'_{\ast }$ is concentrated in degrees $\leq n$, the Eilenberg-MacLane space $\mathrm{K}( A'_{\ast } )$ is $n$-groupoid, which we can identify with the fundamental $n$-groupoid $\pi _{\leq n}(X)$. More precisely, the quotient map $A_{\ast } \twoheadrightarrow A'_{\ast }$ is an isomorphism in degrees $< n$ and induces an isomorphism on homology in degrees $\leq n$, so the induced map of Kan complexes $\mathrm{K}( A_{\ast } ) \twoheadrightarrow \mathrm{K}( A'_{\ast } )$ exhibits $\mathrm{K}(A'_{\ast })$ as a fundamental $n$-groupoid of $X$.

Let $X$ be a Kan complex. Our goal for the remainder of this section is to give an explicit construction of a fundamental $n$-groupoid of $X$, for each $n \geq 0$.

Construction 3.5.6.10. Let $n$ be a nonnegative integer, let $X$ be a Kan complex. and let $\operatorname{cosk}^{\circ }_{n}(X)$ denote the weak $n$-coskeleton of $X$ (Notation 3.5.4.19). For every integer $m \geq 0$, we will identify $m$-simplices of $\operatorname{cosk}^{\circ }_ n(X)$ with morphisms $\sigma : \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow X$ which can be extended to the $(n+1)$-skeleton of $\Delta ^{m}$ (see Remark 3.5.4.21). Given two such morphisms $\sigma , \sigma ': \operatorname{sk}_{n}( \Delta ^ m ) \rightarrow X$, we write $\sigma \sim _{m} \sigma '$ if $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{sk}_{n-1}( \Delta ^{m} )$. The construction

\[ ( [m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{cosk}^{\circ }_{n}(X) ) / \sim _{m} \]

determines a simplicial set, which we will denote by $\pi _{\leq n}(X)$. By construction, we have an epimorphism of simplicial sets $q: \operatorname{cosk}_{n}^{\circ }(X) \twoheadrightarrow \pi _{\leq n}(X)$, which determines a comparison map $X \rightarrow \pi _{\leq n}(X)$.

Remark 3.5.6.11. In the situation of Construction 3.5.6.10, the relation $\sigma \sim _{m} \sigma '$ implies that $\sigma = \sigma '$ whenever $m < n$. It follows that the tautological map $v: X \rightarrow \pi _{\leq n}(X)$ is bijective on simplices of dimension $< n$, and surjective on simplices of dimension $n$.

Proposition 3.5.6.12. Let $X$ be a Kan complex and let $n$ be a nonnegative integer. Then, for every simplicial set $A$, the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \operatorname{cosk}_{n}^{\circ }(X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( A, \pi _{\leq n}(X) ) \]

is surjective. Moreover, if $f_0, f_1: A \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ are morphisms of simplicial sets which correspond to maps $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow X$, then $\theta (f_0) = \theta (f_1)$ if and only if $u_0$ and $u_1$ are homotopic relative to $\operatorname{sk}_{n-1}(A)$.

Proof. We first prove that $\theta $ is a surjection. Fix a morphism $g: A \rightarrow \pi _{\leq n}(X)$. Using Remark 3.5.6.11 (and Proposition 1.1.4.12), we see that $g|_{ \operatorname{sk}_{n}(A) }$ can be lifted to a morphism of simplicial sets $u: \operatorname{sk}_{n}(A) \rightarrow X$. We will show that $u$ can be extended to the $(n+1)$-skeleton of $A$ (and is therefore classified by a morphism $f: A \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ satisfying $\theta (f) = g$; see Remark 3.5.4.21). By virtue of Proposition 1.1.4.12 and Variant 3.2.4.12, this is equivalent to the assertion that for every $(n+1)$-simplex $\sigma $ of $A$ having restriction $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^{n+1} }$, the composition $(g \circ \sigma _0): \operatorname{\partial \Delta }^{n+1} \rightarrow X$ is nullhomotopic. Choose a lift of $g(\sigma )$ to an $(n+1)$-simplex of $\operatorname{cosk}_{n}^{\circ }(X)$, which we identify with a nullhomotopic map $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow X$. By construction, $g \circ \sigma _0$ and $\tau _0$ coincide after composing with the comparison map $v: X \rightarrow \pi _{\leq n}(X)$. Using Proposition 1.1.4.12 again, we see that $g \circ \sigma _0$ and $\tau _0$ are homotopic relative to the $\operatorname{sk}_{n-1}( \Delta ^{n+1} )$, so that $g \circ \sigma _0$ is also nullhomotopic.

Now suppose that we are given a pair of morphisms $f_0, f_1: A \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ satisfying $\theta (f_0) = \theta (f_1)$. We wish to show that the associated maps $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow X$ are homotopic relative to $\operatorname{sk}_{n-1}(A)$ (the converse is immediate from the definitions). Using Remark 3.5.6.11, we deduce that $u_0$ and $u_1$ coincide on $\operatorname{sk}_{n-1}(A)$. By virtue of Proposition 1.1.4.12, we are reduced to showing that for every nondegenerate $n$-simplex $\sigma $ of $A$, the compositions $u_0 \circ \sigma $ and $u_1 \circ \sigma $ are homotopic relative to $\operatorname{\partial \Delta }^{n}$. This follows from our assumption that the maps $\theta (f_0), \theta (f_1): A \rightarrow \pi _{\leq n}(X)$ coincide on the simplex $\sigma $. $\square$

Remark 3.5.6.13. Let $X$ be a Kan complex, let $n \geq 0$ be an integer, and let $A$ be a simplicial set. Stated more informally, Proposition 3.5.6.12 asserts that $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, \pi _{\leq n}(X) )$ can be viewed as a subquotient of the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(A), X)$:

  • A morphism $u: \operatorname{sk}_{n}(A) \rightarrow X$ determines a map from $A$ to $\pi _{\leq n}(X)$ if and only if $u$ can be extended to the $(n+1)$-skeleton of $A$.

  • Two such morphisms $u_0, u_1: \operatorname{sk}_{n}(A) \rightarrow X$ determine the same map from $A$ to $\pi _{\leq n}(X)$ if and only if they are homotopic relative to the $(n-1)$-skeleton of $A$.

Corollary 3.5.6.14. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. Then the comparison map $q: \operatorname{cosk}_{n}^{\circ }(X) \twoheadrightarrow \pi _{\leq n}(X)$ of Construction 3.5.6.10 is a trivial Kan fibration.

Proof. Fix an integer $m \geq 0$; we wish to show that every lifting problem

3.76
\begin{equation} \begin{gathered}\label{equation:coskeleton-vs-fundamental} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ m \ar [d] \ar [r]^-{\sigma _0} & \operatorname{cosk}^{\circ }_{n}(X) \ar [d]^{q} \\ \Delta ^ m \ar@ {-->}[ur] \ar [r]^-{\overline{\sigma }} & \pi _{\leq n}(X) } \end{gathered} \end{equation}

admits a solution.

Let $\sigma $ be any $m$-simplex of $\operatorname{cosk}_{n}^{\circ }(X)$ satisfying $q( \sigma ) = \overline{\sigma }$. By virtue of Remark 3.5.6.11, the commutativity of the diagram (3.76) guarantees that $\sigma _0$ and $\sigma $ coincide on the $(n-1)$-skeleton of $\operatorname{\partial \Delta }^{m}$. Consequently, if $m \leq n$, then $\sigma $ is a solution to the the lifting problem (3.76). We will therefore assume that $m > n$. In this case, the boundary $\operatorname{\partial \Delta }^{m}$ contains the $n$-skeleton of $\Delta ^{m}$. It will therefore suffice to show that $\sigma _0$ can be extended to an $m$-simplex $\sigma '$ of $\operatorname{cosk}_{n}^{\circ }(X)$: the commutativity of the diagram (3.76) guarantees that any such extension satisfies the identity $q( \sigma ' ) = \overline{\sigma }$ (Proposition 3.5.6.12). If $m \geq n+2$, then the existence of $\sigma '$ is automatic (since $\operatorname{cosk}^{\circ }_{n}(X)$ is $(n+1)$-coskeletal). It will therefore suffice to treat the case $m = n+1$. In this case, we can identify $\sigma _0$ with a morphism $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow X$, and we wish to show that $\tau _0$ is nullhomotopic. Note that $\sigma |_{ \operatorname{\partial \Delta }^{m} }$ determines a morphism $\tau _1: \operatorname{\partial \Delta }^{n+1} \rightarrow X$. Moreover, the commutativity of the diagram (3.76) guarantees that $\tau _0$ and $\tau _1$ are homotopic relative to $\operatorname{sk}_{n-1}( \Delta ^{n+1} )$ (Proposition 3.5.6.12). It will therefore suffice to show that $\tau _1$ is nullhomotopic, which follows from the existence of $\sigma $. $\square$

Corollary 3.5.6.15. Let $X$ be a Kan complex, let $n$ be a nonnegative integer, and let $\pi _{\leq n}(X)$ be as in Construction 3.5.6.10. Then the quotient map $\operatorname{cosk}_{n+1}(X) \twoheadrightarrow \pi _{\leq n}(X)$ is a trivial Kan fibration of simplicial sets.

Corollary 3.5.6.16. Let $X$ be a Kan complex, let $n$ be a nonnegative integer, and let $\pi _{\leq n}(X)$ be as in Construction 3.5.6.10. Then $\pi _{\leq n}(X)$ is an $n$-groupoid.

Proof. By virtue of Proposition 3.5.3.23, the coskeleton $\operatorname{cosk}_{n+1}(X)$ is a Kan complex. Combining Corollary3.5.6.15 with Proposition 1.5.5.11, we conclude that $\pi _{\leq n}(X)$ is a Kan complex. To complete the proof, it will suffice to show that if $\sigma $ and $\tau $ are $m$-simplices of $\pi _{\leq n}(X)$ for some $m > n$ which satisfy $\sigma |_{ \Lambda ^{m}_{i} } = \tau |_{ \Lambda ^{m}_{i} }$ for some $0 \leq i \leq m$, then $\sigma = \tau $. Choose maps $\widetilde{\sigma }, \widetilde{\tau }: \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow X$ representing $\sigma $ and $\tau $. Using Proposition 3.5.6.12, we can choose a homotopy from $\widetilde{\sigma }|_{ \operatorname{sk}_{n}( \Lambda ^{m}_{i}) }$ to $\widetilde{\tau }|_{ \operatorname{sk}_{n}(\Lambda ^{m}_{i} ) }$ which is constant when restricted to the skeleton $\operatorname{sk}_{n-1}( \Lambda ^{m}_{i} ) = \operatorname{sk}_{n-1}( \Delta ^ m )$. If $m \geq n+2$, then $h$ is also a homotopy from $\widetilde{\sigma }|_{ \operatorname{sk}_{n}( \Delta ^ m) }$ to $\widetilde{\tau }|_{ \operatorname{sk}_{n}( \Delta ^ m ) }$, so that $\sigma = \tau $ as desired. In the case $m = n+1$, the morphisms $\widetilde{\sigma }$ and $\widetilde{\tau }$ can be extended to morphisms $\overline{\sigma }, \overline{\tau }: \Delta ^{n+1} \rightarrow X$. Using Corollary 3.1.3.6, we can extend $h$ to a homotopy $\overline{h}$ from $\overline{\sigma }$ to $\overline{\tau }$. Restricting this homotopy to the $n$-skeleton of $\Delta ^{m}$, we again conclude that $\sigma = \tau $. $\square$

Theorem 3.5.6.17. Let $X$ be a Kan complex. For every $n \geq 0$, the comparison map $v: X \rightarrow \pi _{\leq n}(X)$ of Construction 3.5.6.10 exhibits $\pi _{\leq n}(X)$ as a fundamental $n$-groupoid of $X$.

Proof. It follows from Remark 3.5.6.11 that $v$ is bijective on $m$-simplices for $m < n$ and surjective on $n$-simplices. By construction, if $\sigma $ and $\sigma '$ are $n$-simplices of $X$, then $v(\sigma ) = v(\sigma ')$ if and only if $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{\partial \Delta }^{n}$. It will therefore suffice to show that $\pi _{\leq n}(X)$ is an $n$-groupoid, which follows from Corollary 3.5.6.16. $\square$