Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

4.3 The Slice and Join Constructions

Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. A cone over $F$ is an object $C \in \operatorname{\mathcal{C}}$ together with a collection of morphisms $\{ \alpha _ K: C \rightarrow F(K) \} _{K \in \operatorname{\mathcal{K}}}$ with the following property: for every morphism $\beta : K \rightarrow K'$ of the category $\operatorname{\mathcal{K}}$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ & C \ar [dl]_{\alpha _ K} \ar [dr]^{\alpha _{K'} } & \\ F(K) \ar [rr]^{ F(\beta ) } & & F(K' ) } \]

commutes. The collection of cones $(C, \{ \alpha _ K \} _{K \in \operatorname{\mathcal{K}}} )$ can be organized into a category, which we will denote by $\operatorname{\mathcal{C}}_{/F}$ and refer to as the slice category of $\operatorname{\mathcal{C}}$ over $F$ (Construction 4.3.1.7). This construction plays an important role in category theory: for example, a limit of the diagram $F$ is (by definition) a final object of the category $\operatorname{\mathcal{C}}_{/F}$.

Our goal in this section is to generalize the construction $(F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{\mathcal{C}}_{/F}$ to the setting of $\infty $-categories. Our first step is to show that the slice category $\operatorname{\mathcal{C}}_{/F}$ can be characterized by a universal property. In §4.3.2, we associate to every pair of categories $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{K}}$ a new category $\operatorname{\mathcal{D}}\star \operatorname{\mathcal{K}}$, which we refer to as the join of $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{K}}$ (Definition 4.3.2.1). This is a new category which contains $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{K}}$ as full subcategories, having a unique morphism from each object of $\operatorname{\mathcal{D}}$ to each object of $\operatorname{\mathcal{K}}$ (and no morphisms in the opposite direction). We then show the datum of a functor $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{/F}$ is equivalent to the datum of a functor $\overline{F}: \operatorname{\mathcal{D}}\star \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{F}|_{\operatorname{\mathcal{K}}} = F$ (Proposition 4.3.2.10).

In §4.3.3, we extend the join construction to the setting of $\infty $-categories. To every pair of simplicial sets $X$ and $Y$, we associate a new simplicial set $X \star Y$ (Construction 4.3.3.13), which contains $X$ and $Y$ as (disjoint) simplicial subsets. This construction has the following features:

  • For every pair of categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, there is a canonical isomorphism of simplicial sets $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \star \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})$ (Example 4.3.3.20). Consequently, the join operation on simplicial sets can be regarded as a generalization of the join operation on categories.

  • For every pair of $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the join $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ is an $\infty $-category (Proposition 4.3.3.21).

  • For every pair of simplicial sets $X$ and $Y$, the join $X \star Y$ is equipped with a continuous bijection

    \[ | X \star Y | \simeq X \coprod _{ (X \times \{ 0\} \times Y) } (X \times [0,1] \times Y) \coprod _{ (X \times \{ 1\} \times Y)} Y, \]

    which is a homeomorphism if either $X$ or $Y$ is finite (Proposition 4.3.4.11 and Corollary 4.3.4.12).

Let $f: K \rightarrow X$ be any morphism of simplicial sets. In §4.3.5, we introduce a new simplicial set $X_{/f}$, which we will refer to as the slice of $X$ over $f$ (Construction 4.3.5.1). The simplicial set $X_{/f}$ is characterized (up to isomorphism) by the following universal mapping property: for any simplicial set $Y$, the datum of a morphism of simplicial sets $Y \rightarrow X_{/f}$ is equivalent to the datum of a morphism of simplicial sets $\overline{f}: Y \star K \rightarrow X$ satisfying $\overline{f}|_{K} = f$ (Proposition 4.3.5.14). Moreover, we will show that it has the following additional properties:

  • If $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between categories and $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{K}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the associated map of simplicial sets, then there is a canonical isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{ / \operatorname{N}_{\bullet }(F)} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/F} )$ (Example 4.3.5.7). Consequently, the slice operation on simplicial sets can be regarded as a generalization of the slice operation on categories.

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $f: K \rightarrow \operatorname{\mathcal{C}}$ is a morphism of simplicial sets, then the simplicial set $\operatorname{\mathcal{C}}_{/f}$ is also an $\infty $-category. Moreover, the evident forgetful functor $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration of $\infty $-categories (Proposition 4.3.6.1).

  • If $q: X \rightarrow S$ is a left fibration of simplicial sets and $f: K \rightarrow X$ is any morphism of simplicial sets, then the natural map $X_{/f} \rightarrow X \times _{S} S_{ / (q \circ f)}$ is a Kan fibration of simplicial sets (Corollary 4.3.7.3).

  • If $q: X \rightarrow S$ is a right fibration of simplicial sets and $x \in X$ is a vertex (which we identify with a map of simplicial sets $\Delta ^{0} \rightarrow X$) having image $s \in S$, then the induced map $X_{/x} \rightarrow S_{/s}$ is a trivial Kan fibration of simplicial sets (Corollary 4.3.7.12).

Structure

  • Subsection 4.3.1: Slices of Categories
  • Subsection 4.3.2: Joins of Categories
  • Subsection 4.3.3: Joins of Simplicial Sets
  • Subsection 4.3.4: Joins of Topological Spaces
  • Subsection 4.3.5: Slices of Simplicial Sets
  • Subsection 4.3.6: Slices of $\infty $-Categories
  • Subsection 4.3.7: Slices of Left and Right Fibrations