Notation 1.2.2.1 (The $n$-Simplex). For each integer $n \geq 0$, we let $| \Delta ^{n} |$ denote the set of $(n+1)$-tuples of nonnegative real numbers $(t_0, t_1, \cdots , t_ n )$ which satisfy the equation $t_0 + t_1 + \cdots + t_ n = 1$. We regard $| \Delta ^{n} |$ as a topological space (with the topology inherited from standard topology on Euclidean space $\mathbf{R}^{n+1}$). If $X$ is a topological space, we will refer to a continuous function $\sigma : | \Delta ^ n | \rightarrow X$ as a singular $n$-simplex in $X$.
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1.2.2 The Singular Simplicial Set of a Topological Space
Topology provides an abundant supply of examples of simplicial sets.
Construction 1.2.2.2. Let $X$ be a topological space. We define a simplicial set $\operatorname{Sing}_{\bullet }(X)$ as follows:
To each object $[n] \in \operatorname{{\bf \Delta }}$, we assign the set $\operatorname{Sing}_{n}(X) = \operatorname{Hom}_{\operatorname{Top}}( | \Delta ^ n |, X )$ of singular $n$-simplices in $X$.
To each non-decreasing map $\alpha : [m] \rightarrow [n]$, we assign the map $\operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{m}(X)$ given by precomposition with the continuous map
\[ | \Delta ^{m} | \rightarrow | \Delta ^{n} | \]
\[ (t_0, t_1, \ldots , t_ m) \mapsto ( \sum _{\alpha (i) = 0} t_ i, \sum _{\alpha (i) = 1} t_ i, \ldots , \sum _{\alpha (i)=n} t_ i). \]
We will refer to $\operatorname{Sing}_{\bullet }(X)$ as the singular simplicial set of $X$. We view the construction $X \mapsto \operatorname{Sing}_{\bullet }(X)$ as a functor from the category of topological spaces to the category of simplicial sets, which we will denote by $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$.
Example 1.2.2.3. Let $X$ be a topological space and let $\operatorname{Sing}_{\bullet }(X)$ be its singular simplicial set. Then:
Vertices of $\operatorname{Sing}_{\bullet }(X)$ can be identified with points of $X$.
Edges of $\operatorname{Sing}_{\bullet }(X)$ can be identified with continuous paths $p: [0,1] \rightarrow X$. Here the source of $p$ is the point $x = p(0)$, and the target of $p$ is the point $y = p(1)$.
Remark 1.2.2.4. The functor $X \mapsto \operatorname{Sing}_{\bullet }(X)$ carries limits in the category of topological spaces to limits in the category of simplicial sets (in fact, the functor $\operatorname{Sing}_{\bullet }$ admits a left adjoint; see Corollary 1.2.3.5). It does not preserve colimits in general. However, it does carry coproducts of topological spaces to coproducts of simplicial sets: this follows from the observation that the topological $n$-simplex $| \Delta ^ n |$ is connected for every $n \geq 0$.
Remark 1.2.2.5 (Connected Components of $\operatorname{Sing}_{\bullet }(X)$). Let $X$ be a topological space. We let $\pi _0(X)$ denote the set of path components of $X$: that is, the quotient of $X$ by the equivalence relation It follows from Remark 1.2.1.23 that we have a canonical bijection $\pi _0( \operatorname{Sing}_{\bullet }(X) ) \simeq \pi _0(X)$. That is, we can identify connected components of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ (in the sense of Definition 1.2.1.8) with path components of the topological space $X$.
\[ (x \sim y ) \Leftrightarrow ( \exists p: [0,1] \rightarrow X) [ p(0) =x \text{ and } p(1) = y]. \]
Remark 1.2.2.6 (Connectedness of $\operatorname{Sing}_{\bullet }(X)$). Let $X$ be a topological space. Then the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is connected if and only if $X$ is path connected (this follows from Remark 1.2.2.5).
Warning 1.2.2.7. Let $X$ be a topological space. If the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is connected, then the topological space $X$ is path connected and therefore connected. Beware that the converse is not necessarily true: there exist topological spaces $X$ which are connected but not path connected, in which case the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ will not be connected.
It will be convenient to consider a generalization of Construction 1.2.2.2.
Variant 1.2.2.8. Let $\operatorname{\mathcal{C}}$ be a category and let $Q$ be a cosimplicial object of $\operatorname{\mathcal{C}}$, which we view as a functor $\operatorname{{\bf \Delta }}$ to $\operatorname{\mathcal{C}}$. For every object $X \in \operatorname{\mathcal{C}}$, the construction $( [n] \in \operatorname{{\bf \Delta }}) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Q( [n] ), X )$ determines a functor from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of sets, which we can view as a simplicial set. We will denote this simplicial set by $\operatorname{Sing}^{Q}_{\bullet }(X)$, so that we have canonical bijections $\operatorname{Sing}^{Q}_{n}(X) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Q^{n}, X)$. We view the construction $X \mapsto \operatorname{Sing}^{Q}_{\bullet }(X)$ as a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets, which we denote by $\operatorname{Sing}^{Q}_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$.
Example 1.2.2.9. The construction $[n] \mapsto | \Delta ^{n} |$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ to the category $\operatorname{Top}$ of topological spaces, which assigns to each morphism $\alpha : [m] \rightarrow [n]$ the continuous map We regard this functor as a cosimplicial topological space, which we denote by $| \Delta ^{\bullet } |$. Applying Variant 1.2.2.8 to this cosimplicial space yields a functor $\operatorname{Sing}^{ | \Delta |}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$, which coincides with the singular simplicial set functor $\operatorname{Sing}_{\bullet }$ of Construction 1.2.2.2.
\[ | \Delta ^{m} | \rightarrow | \Delta ^{n} | \quad \quad (t_0, \ldots , t_ m) \mapsto ( \sum _{\alpha (i) = 0} t_ i, \ldots , \sum _{\alpha (i)=n} t_ i). \]
Example 1.2.2.10. The construction $[n] \mapsto \Delta ^{n}$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ to the category $\operatorname{Set_{\Delta }}= \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set})$ of simplicial sets (this is the Yoneda embedding for the simplex category $\operatorname{{\bf \Delta }}$). We regard this functor as a cosimplicial object of $\operatorname{Set_{\Delta }}$, which we denote by $\Delta ^{\bullet }$. Applying Variant 1.2.2.8 to this cosimplicial object, we obtain a functor from the category of simplicial sets to itself, which is canonically isomorphic to the identity functor $\operatorname{id}_{ \operatorname{Set_{\Delta }}}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ (see Proposition 1.1.0.12).
Remark 1.2.2.11. The cosimplicial space $| \Delta ^{\bullet } |$ of Example 1.2.2.9 can be described more informally as follows:
To each nonempty finite linearly ordered set $I$, it assigns a topological simplex $| \Delta ^{I} |$ whose vertices are the elements of $I$: that is, the convex hull of the set $I$ inside the real vector space $\operatorname{\mathbf{R}}[I]$ generated by $I$.
To every nondecreasing map $\alpha : I \rightarrow J$, the induced map $| \Delta ^{I} | \rightarrow | \Delta ^{J} |$ is given by the restriction of the $\operatorname{\mathbf{R}}$-linear map $\operatorname{\mathbf{R}}[I] \rightarrow \operatorname{\mathbf{R}}[J]$ determined by $\alpha $. Equivalently, it is the unique affine map which coincides with $\alpha $ on the vertices of the simplex $| \Delta ^{I} |$.