Definition 1.2.1.1. Let $S_{}$ be a simplicial set and let $S'_{} \subseteq S_{}$ be a simplicial subset of $S_{}$ (Remark 1.1.0.14). We will say that $S'_{}$ is a summand of $S_{}$ if the simplicial set $S_{}$ decomposes as a coproduct $S'_{} \coprod S''_{}$, for some other simplicial subset $S''_{} \subseteq S_{}$.
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1.2.1 Connected Components of Simplicial Sets
In this section, we introduce the notion of a connected simplicial set (Definition 1.2.1.6 and show that every simplicial set $S_{}$ decomposes uniquely as a disjoint union of connected subsets (Proposition 1.2.1.13), indexed by a set $\pi _0( S_{} )$ which we call the set of connected components of $S_{}$. Moreover, we characterize the construction $S_{} \mapsto \pi _0( S_{} )$ as a left adjoint to the functor $I \mapsto \underline{I}_{}$ of Construction 1.1.5.2 (Corollary 1.2.1.21).
Remark 1.2.1.2. In the situation of Definition 1.2.1.1, if $S'_{\bullet } \subseteq S_{\bullet }$ is a summand, then the complementary summand $S''_{\bullet }$ is uniquely determined: for each $n \geq 0$, we must have $S''_{n} = S_{n} \setminus S'_{n}$. Consequently, the condition that $S'_{\bullet }$ is a summand of $S_{\bullet }$ is equivalent to the condition that the construction is functorial: that is, that the face and degeneracy operators for the simplicial set $S_{\bullet }$ preserve the subsets $S_{n} \setminus S'_{n}$.
\[ ( [n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto S_{n} \setminus S'_{n} \]
Remark 1.2.1.3. Let $S_{}$ be a simplicial set. Then the collection of all summands of $S_{}$ is closed under the formation of unions and intersections (this follows immediately from the criterion of Remark 1.2.1.2).
Remark 1.2.1.4 (Transitivity). Let $S_{}$ be a simplicial set. If $S'_{} \subseteq S_{}$ is a summand of $S_{}$ and $S''_{} \subseteq S'_{}$ is a summand of $S'_{}$, then $S''_{}$ is a summand of $S_{}$.
Remark 1.2.1.5. Let $f: S_{} \rightarrow T_{}$ be a map of simplicial sets and let $T'_{} \subseteq T_{}$ be a summand. Then the inverse image $f^{-1}( T'_{} ) \simeq S_{} \times _{ T_{} } T'_{}$ is a summand of $S_{}$.
Definition 1.2.1.6. Let $S_{}$ be a simplicial set. We will say that $S_{}$ is connected if it is nonempty and every summand $S'_{} \subseteq S_{}$ is either empty or coincides with $S_{}$.
Example 1.2.1.7. For each $n \geq 0$, the standard $n$-simplex $\Delta ^{n}$ is connected.
Definition 1.2.1.8 (Connected Components). Let $S_{}$ be a simplicial set. We will say that a simplicial subset $S'_{} \subseteq S_{}$ is a connected component of $S_{}$ if $S'_{}$ is a summand of $S_{}$ (Definition 1.2.1.1) and $S'_{}$ is connected (Definition 1.2.1.6). We let $\pi _0( S_{} )$ denote the set of all connected components of $S_{}$.
Warning 1.2.1.9. Let $S$ be a simplicial set. As we will soon see, the set $\pi _0(S_{})$ admits many different descriptions:
We can identify $\pi _0( S_{} )$ with the set of connected components of $S_{}$ (Definition 1.2.1.8).
We can identify $\pi _0( S_{} )$ with a colimit of the diagram $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ given by the simplicial set $S_{}$ (Remark 1.2.1.20).
We can identify $\pi _0( S_{} )$ with the quotient of the set of vertices of $S$ by an equivalence relation $\sim $ generated by the set of edges of $S$ (Remark 1.2.1.23).
We can identify $\pi _0( S_{} )$ with the set of connected components of the directed graph $\mathrm{Gr}( S_{} )$ introduced in ยง1.1.6 (Variant 1.2.1.24).
If $S_{}$ is a Kan complex, we can identify $\pi _0( S )$ as the set of isomorphism classes of objects in the fundamental groupoid $\pi _{\leq 1}(S_{})$ (Remark 1.4.6.13).
Because of this abundance of perspectives, it often will be convenient to view $I = \pi _0( S_{} )$ as an abstract index set which is equipped with a bijection
\[ I \simeq \{ \text{Connected components of $S_{}$} \} \quad \quad (i \in I) \mapsto (S'_{i } \subseteq S_{}), \]
rather than as the set of connected components itself.
Example 1.2.1.10. Let $I$ be a set and let $\underline{I}_{}$ be the constant simplicial set associated to $I$ (Construction 1.1.5.2). Then the connected components of $\underline{I}_{}$ are exactly the simplicial subsets of the form $\underline{ \{ i \} }$ for $i \in I$. In particular, we have a canonical bijection $I \simeq \pi _0( \underline{I}_{} )$.
Proposition 1.2.1.11. Let $f: S_{} \rightarrow T_{}$ be a map of simplicial sets, and suppose that $S_{}$ is connected. Then there is a unique connected component $T'_{} \subseteq T_{}$ such that $f( S_{} ) \subseteq T'_{}$.
Proof. Let $T'_{}$ be the smallest summand of $T_{}$ which contains the image of $f$ (the existence of $T'_{}$ follows from Remark 1.2.1.3: we can take $T'_{}$ to be the intersection of all those summands of $T_{}$ which contain the image of $f$). We will complete the proof by showing that $T'_{}$ is connected. Since $S_{}$ is nonempty, $T'_{}$ must be nonempty. Let $T''_{} \subseteq T'_{}$ be a summand; we wish to show that $T''_{} = T'_{}$ or $T''_{} = \emptyset $. Note that $f^{-1}(T''_{} )$ is a summand of $S_{}$ (Remark 1.2.1.5). Since $S_{}$ is connected, we must have $f^{-1}( T''_{} ) = S_{}$ or $f^{-1}( T''_{} ) = \emptyset $. Replacing $T''_{}$ by its complement if necessary, we may assume that $f^{-1}( T''_{} ) = S_{}$, so that $f$ factors through $T''_{}$. Since $T''_{}$ is a summand of $T_{}$ (Remark 1.2.1.4), the minimality of $T'_{}$ guarantees that $T''_{} = T'_{}$, as desired. $\square$
Corollary 1.2.1.12. Let $S_{}$ be a simplicial set. The following conditions are equivalent:
The simplicial set $S_{}$ is connected.
For every set $I$, the canonical map
is bijective.
\[ I \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^0, \underline{I}_{} ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{}, \underline{I}_{} ) \]
Proof. The implication $(a) \Rightarrow (b)$ follows from Proposition 1.2.1.11 and Example 1.2.1.10. Conversely, suppose that $(b)$ is satisfied. Applying $(b)$ in the case $I = \emptyset $, we conclude that there are no maps from $S_{}$ to the empty simplicial set, so that $S_{}$ is nonempty. If $S_{}$ is a disjoint union of simplicial subsets $S'_{}, S''_{} \subseteq S_{}$, then we obtain a map of simplicial sets
\[ S_{} \simeq S'_{} \coprod S''_{} \rightarrow \Delta ^0 \coprod \Delta ^0 \]
and assumption $(b)$ guarantees that this map factors through one of the summands on the right hand side; it follows that either $S'_{}$ or $S''_{}$ is empty. $\square$
Proposition 1.2.1.13. Let $S_{}$ be a simplicial set. Then $S_{}$ is the disjoint union of its connected components.
Proof. Let $\sigma $ be an $n$-simplex of $S_{}$; we wish to show that there is a unique connected component of $S_{}$ which contains $\sigma $. This follows from Proposition 1.2.1.11, applied to the map $\Delta ^ n \rightarrow S_{}$ classified by $\sigma $ (since the standard $n$-simplex $\Delta ^ n$ is connected; see Example 1.2.1.7). $\square$
Corollary 1.2.1.14. Let $S_{}$ be a simplicial set. Then $S_{}$ is empty if and only if $\pi _0( S_{} )$ is empty.
Corollary 1.2.1.15. Let $S_{}$ be a simplicial set. Then $S_{}$ is connected if and only if $\pi _0( S_{})$ has exactly one element.
Exercise 1.2.1.16 (Classification of Summands). Let $S_{}$ be a simplicial set. Show that a simplicial subset $S'_{} \subseteq S_{}$ is a summand if and only if it can be written as a union of connected components of $S_{}$. Consequently, we have a canonical bijection
\[ \{ \text{Subsets of $\pi _0(S_{})$} \} \simeq \{ \text{Summands of $S_{}$} \} . \]
Remark 1.2.1.17 (Functoriality of $\pi _0$). Let $f: S_{} \rightarrow T_{}$ be a map of simplicial sets. It follows from Proposition 1.2.1.11 that for each connected component $S'_{} \subseteq S_{}$, there is a unique connected component $T'_{} \subseteq T_{}$ such that $f(S'_{} ) \subseteq T'_{}$. The construction $S'_{} \mapsto T'_{}$ then determines a map of sets $\pi _0(f): \pi _0(S_{}) \rightarrow \pi _{0}( T_{} )$. This construction is compatible with composition, and therefore allows us to view the construction $S_{} \mapsto \pi _0( S_{} )$ as a functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ from the category of simplicial sets to the category of sets.
We now show that the connected component functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ can be characterized by a universal property.
Construction 1.2.1.18 (The Component Map). Let $S_{}$ be a simplicial set. For every $n$-simplex $\sigma $ of $S_{}$, Proposition 1.2.1.13 implies that there is a unique connected component $S'_{} \subseteq S_{}$ which contains $\sigma $. The construction $\sigma \mapsto S'_{}$ then determines a map of simplicial sets where $\underline{ \pi _0( S_{})}$ denotes the constant simplicial set associated to $\pi _0(S_{})$ (Construction 1.1.5.2). We will refer to $u$ as the component map.
\[ u: S_{} \rightarrow \underline{ \pi _0( S_{})}, \]
Proposition 1.2.1.19. Let $S_{}$ be a simplicial set and let $u: S_{} \rightarrow \underline{ \pi _0( S_{})}_{}$ be the component map of Construction 1.2.1.18. For every set $J$, composition with $u$ induces a bijection
\[ \operatorname{Hom}_{\operatorname{Set}}( \pi _0(S_{}), J ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{}, \underline{J}_{} ). \]
Proof. Decomposing $S_{}$ as the union of its connected components, we can reduce to the case where $S_{}$ is connected, in which case the desired result is a reformulation of Corollary 1.2.1.12. $\square$
Remark 1.2.1.20 ($\pi _0$ as a Colimit). Let $S_{}$ be a simplicial set. It follows from Proposition 1.2.1.19 that the component map $u: S_{} \rightarrow \underline{ \pi _0( S_{})}_{}$ exhibits $\pi _0( S_{} )$ as the colimit of the diagram $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ determined by $S_{}$.
Corollary 1.2.1.21. The connected component functor of Remark 1.2.1.17 is left adjoint to the constant simplicial set functor of Construction 1.1.5.2. More precisely, the construction $S_{} \mapsto (u: S_{} \rightarrow \underline{ \pi _0( S_{})}_{})$ is the unit of an adjunction.
\[ \pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad S_{} \mapsto \pi _0( S_{} ) \]
\[ \operatorname{Set}\rightarrow \operatorname{Set_{\Delta }}\quad \quad I \mapsto \underline{I}_{} \]
We now make Remark 1.2.1.20 more concrete.
Proposition 1.2.1.22. Let $S_{\bullet }$ be a simplicial set, and let $u_0: S_0 \rightarrow \pi _0( S_{\bullet } )$ be the map of sets given by the component map of Construction 1.2.1.18. Then $u_0$ exhibits $\pi _0(S_{\bullet } )$ as the coequalizer of the face operators $d^{1}_0, d^{1}_1: S_{1} \rightrightarrows S_0$.
Remark 1.2.1.23. Let $S_{\bullet }$ be a simplicial set. Proposition 1.2.1.22 supplies a coequalizer diagram of sets In other words, it allows us to identify $\pi _0( S_{\bullet } )$ with the quotient of $S_0 / \sim $, where $\sim $ is the equivalence relation generated by the set of edges of $S_{\bullet }$ (that is, the smallest equivalence relation with the property that $d^{1}_0(e) \sim d^{1}_1(e)$, for every edge $e \in S_1$). In particular, the set $\pi _0( S_{\bullet } )$ depends only on the $1$-skeleton of $S_{\bullet }$.
\[ \xymatrix@R =50pt@C=50pt{ S_1 \ar@ <.4ex>[r]^-{d^{1}_0} \ar@ <-.4ex>[r]_-{d^{1}_1} & S_0 \ar [r] & \pi _0(S_{\bullet }).} \]
Variant 1.2.1.24. Let $S_{\bullet }$ be a simplicial set. Then the set of connected components $\pi _0( S_{\bullet } )$ can also be described as the coequalizer of the pair of maps $d^{1}_0, d^{1}_1: S_{1}^{ \mathrm{nd} } \rightrightarrows S_0$, where $S_{1}^{\mathrm{nd} } \subseteq S_{1}$ denotes the set of nondegenerate edges of $S_{\bullet }$ (since every degenerate edge $e \in S_{1}$ automatically satisfies $d^{1}_0(e) = d^{1}_1(e)$). We therefore have a coequalizer diagram of sets where $G = \mathrm{Gr}(S_{\bullet } )$ is the directed graph of Example 1.1.6.4. In other words, we can identify $\pi _0( S_{\bullet } )$ with the set of connected components of $G$, in the usual graph-theoretic sense.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Edge}(G) \ar@ <.4ex>[r]^-{s} \ar@ <-.4ex>[r]_-{t} & \operatorname{Vert}(G) \ar [r] & \pi _0(S_{\bullet }),} \]
Corollary 1.2.1.25. For $n \geq 2$, the simplicial set $\operatorname{\partial \Delta }^{n}$ is connected.
Proof. Example 1.2.1.7 guarantees that the standard simplex $\Delta ^ n$ is connected. The desired result now follows from Proposition 1.2.1.22, since the inclusion map $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ is bijective on simplices of dimension $\leq 1$. $\square$
Proof of Proposition 1.2.1.22. Let $I$ be a set and let $f: S_0 \rightarrow I$ be a function satisfying $f \circ d^{1}_0 = f \circ d^{1}_1$ (as functions from $S_1$ to $I$). We wish to show that $f$ factors uniquely as a composition
\[ S_0 \xrightarrow {u_0} \pi _0(S_{\bullet } ) \rightarrow I. \]
By virtue of Proposition 1.2.1.19, this is equivalent to the assertion that there is a unique map of simplicial sets $F: S_{\bullet } \rightarrow \underline{I}_{}$ which coincides with $f$ on simplices of degree zero. Let $\sigma $ be an $n$-simplex of $S_{\bullet }$, which we identify with a map of simplicial sets $\sigma : \Delta ^ n \rightarrow S_{\bullet }$. For $0 \leq i \leq n$, we regard $\sigma (i)$ as a vertex of $S_{\bullet }$. Note that if $0 \leq i \leq j \leq n$, then we have $f( \sigma (i) ) = f ( \sigma (j) )$: to prove this, we can assume without loss of generality that $i=0$ and $j = n = 1$, in which case it follows from our hypothesis that $f \circ d^{1}_0 = f \circ d^{1}_1$. It follows that there is a unique element $F(\sigma ) \in I$ such that $F(\sigma ) = f( \sigma (i) )$ for each $0 \leq i \leq n$. The construction $\sigma \mapsto F(\sigma )$ defines a map of simplicial sets $F: S_{\bullet } \rightarrow \underline{I}_{}$ with the desired properties. $\square$
Proposition 1.2.1.26. The collection of connected simplicial sets is closed under finite products.
Proof. Since the final object $\Delta ^{0} \in \operatorname{Set_{\Delta }}$ is connected (Example 1.2.1.7), it will suffice to show that the collection of connected simplicial sets is closed under pairwise products. Let $S_{\bullet }$ and $T_{\bullet }$ be connected simplicial sets; we wish to show that $S_{} \times T_{}$ is connected. Equivalently, we wish to show that $\pi _0( S_{\bullet } \times T_{\bullet } )$ consists of a single element (Corollary 1.2.1.15). By virtue of Proposition 1.2.1.22, the component map supplies a surjection
\[ u_0: S_0 \times T_0 \twoheadrightarrow \pi _0( S_{\bullet } \times T_{\bullet } ). \]
It will therefore suffice to show that every pair of vertices $(s,t), (s', t') \in S_0 \times T_0$ belong to the same connected component of $S_{\bullet } \times T_{\bullet }$. Let $K_{\bullet } \subseteq S_{\bullet } \times T_{\bullet }$ be the connected component which contains the vertex $(s', t )$. Since $S_{\bullet }$ is connected, the map
\[ S_{\bullet } \simeq S_{\bullet } \times \{ t \} \hookrightarrow S_{\bullet } \times T_{\bullet } \]
factors through a unique connected component of $S_{\bullet } \times T_{\bullet }$, which must be equal to $K_{\bullet }$. It follows that $K_{\bullet }$ contains the vertex $(s,t)$. A similar argument (with the roles of $S_{\bullet }$ and $T_{\bullet }$ reversed) shows that $K_{\bullet }$ contains $(s',t')$. $\square$
Corollary 1.2.1.27. The functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ preserves finite products.
Proof. Since $\pi _0( \Delta ^0)$ is a singleton (Example 1.2.1.7), it will suffice to show that for every pair of simplicial sets $S_{\bullet }$ and $T_{\bullet }$, the canonical map
\[ \pi _0( S_{\bullet } \times T_{\bullet } ) \rightarrow \pi _0( S_{\bullet } ) \times \pi _0( T_{\bullet } ) \]
is bijective. Writing $S_{\bullet }$ and $T_{\bullet }$ as a disjoint union of connected components (Proposition 1.2.1.13), we can reduce to the case where $S_{\bullet }$ and $T_{\bullet }$ are connected, in which case the desired result follows from Proposition 1.2.1.26. $\square$
Warning 1.2.1.28. The collection of connected simplicial sets is not closed under infinite products (so the functor $\pi _0: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}$ does not commute with infinite products). For example, let $G$ be the directed graph with vertex set $\operatorname{Vert}(G) = \operatorname{\mathbf{Z}}_{\geq 0} = \operatorname{Edge}(G)$, with source and target maps More informally, $G$ is the directed graph depicted in the diagram The associated $1$-dimensional simplicial set $G_{\bullet }$ is connected. However, the infinite product $S_{\bullet } = \prod _{n \in \operatorname{\mathbf{Z}}_{\geq 0}} G_{\bullet }$ is not connected. By definition, the vertices of $S_{\bullet }$ can be identified with functions $f: \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathbf{Z}}_{\geq 0}$. It is not difficult to see that two such functions $f,g: \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathbf{Z}}_{\geq 0}$ belong to the same connected component of $S_{\bullet }$ if and only if the function $n \mapsto | f(n) - g(n) |$ is bounded. In particular, the identity function $n \mapsto n$ and the zero function $n \mapsto 0$ do not belong to the same connected component of $S_{\bullet }$.
\[ s,t: \operatorname{Edge}(G) \rightarrow \operatorname{Vert}(G) \quad \quad s(n) = n \quad \quad t(n) = n+1. \]
\[ \xymatrix@R =50pt@C=50pt{ 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & 3 \ar [r] & 4 \ar [r] & \cdots } \]