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10.1 Simplicial Objects of $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be a category. Recall that a simplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{{\bf \Delta }}$ is the simplex category introduced in Definition 1.1.0.2. This notion has an obvious counterpart in the setting of $\infty $-categories:

Definition 10.1.0.1 (Simplicial Objects). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A simplicial object of $\operatorname{\mathcal{C}}$ is a functor from the $\infty $-category $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$ to $\operatorname{\mathcal{C}}$. A cosimplicial object of $\operatorname{\mathcal{C}}$ is a functor from $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})$ to $\operatorname{\mathcal{C}}$.

Notation 10.1.0.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will often use the notation $X_{\bullet }$ to indicate a 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 an expression like $X^{\bullet }$ to indicate a cosimplicial object of $\operatorname{\mathcal{C}}$, and $X^{n}$ for its value on the object $[n] \in \operatorname{{\bf \Delta }}$.

Example 10.1.0.3. Let $\operatorname{\mathcal{C}}$ be a category. Then (co)simplicial objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.1.0.1) can be identified with (co)simplicial objects of $\operatorname{\mathcal{C}}$ (in the sense of Definition 1.1.0.4).

Notation 10.1.0.4 (Face and Degeneracy Operators). Let $X_{\bullet }$ be a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. For every pair of integers $0 \leq i \leq n$, we let $s^{n}_{i}: X_{n} \rightarrow X_{n+1}$ denote the morphism induced by the surjection $\sigma ^{i}_{n}: [n+1] \twoheadrightarrow [n]$ of Construction 1.1.2.1; we will refer to $s^{n}_{i}$ as the $i$th degeneracy operator for the simplicial object $X_{\bullet }$. If $n > 0$, we let $d^{n}_{i}: X_{n} \rightarrow X_{n-1}$ denote the morphism induced by the inclusion of linearly ordered sets $\delta ^{i}_{n}: [n-1] \hookrightarrow [n]$ introduced in Construction 1.1.1.4. We will refer to $d^{n}_{i}$ as the $i$th face operator of $X_{\bullet }$.

Warning 10.1.0.5. If $\operatorname{\mathcal{C}}$ is an ordinary category, then a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is completely determined by the collection of objects $\{ X_{n} \} _{n \geq 0}$, together with the face and degeneracy operators

\[ d^{n}_{i}: X_{n} \rightarrow X_{n-1} \quad \quad s^{n}_{i}: X_{n} \rightarrow X_{n+1} \]

(see Proposition 1.1.2.14). In the setting of $\infty $-categories, this is no longer true.

Structure

  • Subsection 10.1.1: Geometric Realization
  • Subsection 10.1.2: Semisimplicial Objects
  • Subsection 10.1.3: Skeletal Simplicial Objects
  • Subsection 10.1.4: Coskeletal Simplicial Objects
  • Subsection 10.1.5: The ČechNerve of a Morphism
  • Subsection 10.1.6: Split Simplicial Objects