Let $S$ be a simplicial set. Recall that the geometric realization of $S$ is a topological space $|S|$ which corepresents the functor
here $\operatorname{Top}$ denotes the category whose objects are topological spaces and whose morphisms are continuous functions (Definition 1.2.3.1). This property determines the topological space $| S |$ up to homeomorphism: that is, up to isomorphism in the category $\operatorname{Top}$. We now formulate a homotopy-invariant counterpart of this universal property, which determines the topological space $| S |$ up to homotopy equivalence (rather than homeomorphism). In what follows, we regard $\operatorname{Top}$ as a simplicially enriched category (see Example 2.4.1.5), and we let $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$ denote its homotopy coherent nerve.
Proposition 10.1.1.1. Let $S$ be a simplicial set. Then the geometric realization $| S |$ is a colimit of the diagram
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { S } \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top}). \]
Proposition 10.1.1.1 admits a more combinatorial formulation:
Variant 10.1.1.2. Let $S = S_{\bullet }$ be a simplicial set. Then the Kan complex $\operatorname{Sing}_{\bullet }( | S | )$ is a colimit of the diagram
\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { S } \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}}. \]
Proof of Variant 10.1.1.2. Let $\operatorname{Kan}$ denote the ordinary category of Kan complexes, and let $\mathscr {F}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Kan}$ be the functor which carries each object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the set of $n$-simplices $S_{n}$ (regarded as a constant simplicial set). Let $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ denote the homotopy colimit of the diagram $\mathscr {F}$ (Construction 5.3.2.1). By virtue of Proposition 7.5.7.1 (and Example 1.4.2.5), it will suffice to show that there is a weak homotopy equivalence of simplicial sets $v: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )^{\operatorname{op}} \rightarrow \operatorname{Sing}_{\bullet }( | S | )$. Using Example 5.3.2.5 (and Example 5.2.6.4), we can identify $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )^{\operatorname{op}}$ with the nerve of the category of simplices $\operatorname{{\bf \Delta }}_{S}$ (see Construction 1.1.3.9). We complete the proof by taking $v$ to be the composition
where $u$ is the weak homotopy equivalence of Theorem 3.6.4.1 and $\psi : \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{S} ) \rightarrow S$ is the comparison map of Construction 3.3.3.9. By virtue of Variant 6.3.7.4, the morphism $\psi $ is universally localizing, and is therefore also a weak homotopy equivalence (Remark 6.3.6.5). $\square$
Proof of Proposition 10.1.1.1. Let $\operatorname{\mathcal{T}}$ be the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$, and let $\operatorname{\mathcal{T}}_{0} \subseteq \operatorname{\mathcal{T}}$ be the full subcategory spanned by those topological spaces which have the homotopy type of a CW complex. It follows from Example 6.2.2.7 that $\operatorname{\mathcal{T}}_0$ is a coreflective subcategory of $\operatorname{\mathcal{T}}$; in particular, the inclusion map $\operatorname{\mathcal{T}}_0 \hookrightarrow \operatorname{\mathcal{T}}$ preserves colimits (Variant 7.1.3.24). It will therefore suffice to show that for every simplicial set $S$, the geometric realization $| S |$ is a colimit of the diagram $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \xrightarrow { S } \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{T}}_{0}$. This is a reformulation of Variant 10.1.1.2, since the functor $X \mapsto \operatorname{Sing}_{\bullet }(X)$ determines an equivalence of $\infty $-categories $\operatorname{\mathcal{T}}_{0} \rightarrow \operatorname{\mathcal{S}}$ (Remark 5.5.1.9). $\square$
Motivated by Proposition 10.1.1.1, we introduce the following terminology:
Definition 10.1.1.3 (Geometric Realization). Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. We will say that an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ if it is a colimit of the diagram $X_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$.
Warning 10.1.1.4. Let $S = S_{\bullet }$ be a simplicial set. Proposition 10.1.1.1 asserts that the topological space $| S |$ introduced in §1.2.3 is a geometric realization of $S$ (in the sense of Definition 10.1.1.3), provided that we regard $S$ as a simplicial object of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Top})$ (by equipping each of the sets $S_{n}$ with the discrete topology). Beware that $| S_{\bullet } |$ is usually not a geometric realization of $S$ (in the sense of Definition 10.1.1.3) if we regard $S$ as a simplicial object of the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{Set})$. The latter is a colimit of the diagram $\operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { S } \operatorname{Set}$, which identifies with the set of connected components $\pi _0( S )$ (see Remark 1.2.1.20), or equivalently with the set of path components of the topological space $|S|$ (see Corollary 1.2.3.19).
Notation 10.1.1.5. Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. It follows from Proposition 7.1.1.12 that, if $X_{\bullet }$ admits a geometric realization $X$, then the isomorphism class of $X$ is uniquely determined. To emphasize this, we will often denote $X$ by $| X_{\bullet } |$ and refer to it as the geometric realization of $X_{\bullet }$. Beware that, in the case where $\operatorname{\mathcal{C}}$ is (the nerve of) the category of sets, this is incompatible with the convention of Notation 1.2.3.3 (see Warning 10.1.1.4).
Exercise 10.1.1.6. Let $X_{\bullet }$ be a simplicial object of an ordinary category $\operatorname{\mathcal{C}}$. Show that an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ (in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$) if and only if it is a coequalizer of the face operators $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$. For a slightly more general statement, see Corollary 10.1.2.12.
Example 10.1.1.7 (Simplicial Abelian Groups). Let $\operatorname{ Ab }$ denote the category of abelian groups. By virtue of the Dold-Kan correspondence (Theorem 2.5.6.1), there is an equivalence of categories $\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{ Ab }) \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})_{\geq 0}$, which carries each simplicial abelian group $A_{\bullet }$ to its normalized Moore complex Under this equivalence, the coequalizer of the pair of face operators $d^{1}_0, d^{1}_1: A_1 \rightrightarrows A_0$ can be identified with the $0$th homology group $\mathrm{H}_{0}( \mathrm{N}_{\ast }(A) ) = \operatorname{coker}( \partial : \mathrm{N}_{1}(A) \rightarrow \mathrm{N}_{0}(A) )$, or alternatively with the homotopy group $\pi _0( A_{\bullet } )$ (see Exercise 3.2.2.22). Using Exercise 10.1.1.6 we see that the $\pi _0(A)$ can be regarded as a geometric realization of $A_{\bullet }$ in the category of abelian groups. In particular, the forgetful functor $\operatorname{ Ab }\rightarrow \operatorname{Set}$ commutes with the formation of geometric realizations (this is a special case of a more general phenomenon, which we will return to in §
\[ \mathrm{N}_{\ast }(A) = ( \cdots \rightarrow \mathrm{N}_{2}(A) \xrightarrow {\partial } \mathrm{N}_{1}(A) \xrightarrow {\partial } \mathrm{N}_{0}(A) ). \]
).
Remark 10.1.1.8. Let $X_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. It follows from Exercise 10.1.1.6 that a geometric realization of $X_{\bullet }$ (if it exists) depends only on the pair of face operators $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$. Beware that, in the $\infty $-categorical setting, this is generally not true: the geometric realization $| X_{\bullet } |$ is sensitive to information about the entire simplicial object $X_{\bullet }$.
Variant 10.1.1.9. Let $X^{\bullet }$ be a cosimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. We will say that an object $X \in \operatorname{\mathcal{C}}$ is a totalization of $X^{\bullet }$ if it is a limit of the diagram $X^{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) \rightarrow \operatorname{\mathcal{C}}$. If this condition is satisfied, then $X$ is uniquely determined up to isomorphism. To emphasize this, we will often denote $X$ by $\operatorname{Tot}( X^{\bullet } )$ and refer to it as the totalization of $X^{\bullet }$.
For many applications, the language of Definition 10.1.1.3 is insufficiently precise. Given a simplicial object $X_{\bullet }$ of an $\infty $-category $\operatorname{\mathcal{C}}$, we would like to view its geometric realization $| X_{\bullet } |$ not abstractly as an object of $\operatorname{\mathcal{C}}$, but as an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ X_{\bullet } / }$. For this purpose, it will be convenient to introduce some additional terminology.
Definition 10.1.1.10 (The Augmented Simplex Category). For each integer $n \geq -1$, let $[n]$ denote the linearly ordered set $\{ 0 < 1 < \cdots < n \} $, so that $[-1]$ is the empty set. We let $\operatorname{{\bf \Delta }}_{+}$ denote the category whose objects are the linearly ordered sets $\{ [n] \} _{n \geq -1}$, and whose morphisms are nondecreasing functions. We will refer to $\operatorname{{\bf \Delta }}_{+}$ as the augmented simplex category.
Remark 10.1.1.11. The augmented simplex category $\operatorname{{\bf \Delta }}_{+}$ of Definition 10.1.1.10 contains the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.0.2 as a full subcategory (spanned by the objects $[n]$ for $n \geq 0$). Moreover, $\operatorname{{\bf \Delta }}_{+}$ can be obtained from $\operatorname{{\bf \Delta }}$ by adjoining a single object $[-1]$, which is an initial object satisfying $\operatorname{Hom}_{ \operatorname{{\bf \Delta }}_{+} }( [n], [-1] ) = \emptyset $ for $n \geq 0$. In other words, $\operatorname{{\bf \Delta }}_{+}$ can be identified with the left cone $\operatorname{{\bf \Delta }}^{\triangleleft }$ (see Example 4.3.2.5).
Definition 10.1.1.12 (Augmented Simplicial Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An augmented simplicial object of $\operatorname{\mathcal{C}}$ is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} )$ to $\operatorname{\mathcal{C}}$. An augmented cosimplicial object is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+} )$ to $\operatorname{\mathcal{C}}$.
Notation 10.1.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often use the notation $X_{\bullet }$ to indicate an augmented simplicial object of $\operatorname{\mathcal{C}}$. In this case, we write $X_{n}$ for the value of the functor $X_{\bullet }$ on the object $[n] \in \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}$. Similarly, we often use the expression $X^{\bullet }$ to indicate an augmented cosimplicial object of $\operatorname{\mathcal{C}}$, and $X^{n}$ for its value on the object $[n] \in \operatorname{{\bf \Delta }}_{+}$.
Remark 10.1.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Every augmented simplicial object of $\operatorname{\mathcal{C}}$ determines a simplicial object of $\operatorname{\mathcal{C}}$, by restriction along the inclusion of full subcategories $\operatorname{{\bf \Delta }}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+}$. For this reason, we will sometimes use the notation $\overline{X}_{\bullet }$ to indicate an augmented simplicial object of $\operatorname{\mathcal{C}}$, to distinguish it from the underlying simplicial object $X_{\bullet } = \overline{X}_{\bullet }|_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )}$.
Remark 10.1.1.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. By virtue of Remark 10.1.1.11, the following data are equivalent:
Augmented simplicial objects $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}$ carrying the object $[-1]$ to $X$.
Simplicial objects of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$.
We will often invoke this equivalence implicitly, using the notation $X_{\bullet }$ to indicate both an augmented simplicial object of $\operatorname{\mathcal{C}}$ (satisfying $X_{-1} = X$) and the associated simplicial object of $\operatorname{\mathcal{C}}_{/X}$.
Definition 10.1.1.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$, let $\overline{X}_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$ satisfying $\overline{X}_{-1} = X$, and let $X_{\bullet } = \underline{X}_{\bullet } |_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) }$ denote its underlying simplicial object. We will say that $\overline{X}_{\bullet }$ exhibits $X$ as a geometric realization of $X_{\bullet }$ if it is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, in the sense of Variant 7.1.2.5. Similarly, if $\overline{X}^{\bullet }$ is an augmented cosimplicial object of $\operatorname{\mathcal{C}}$ satisfying $\overline{X}^{-1} = X$ and $X^{\bullet } = \overline{X}^{\bullet }|_{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) }$ is the underlying cosimplicial object, we say that $\overline{X}^{\bullet }$ exhibits $X$ as a totalization of $X^{\bullet }$ if it is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.2.4.
Remark 10.1.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then $X$ is a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.1.1.3) if and only if there exists an augmented simplicial object $\overline{X}_{\bullet }$ which exhibits $X$ as a geometric realization of $X_{\bullet }$ (in the sense of Definition 10.1.1.16). See Remark 7.1.2.7.