Kerodon

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

8.1 Twisted Arrows and Cospans

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §4.6.1, we associated to every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ a Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, whose vertices are morphisms from $X$ to $Y$. In §, we will see that the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be refined to a functor of $\infty $-categories

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , \bullet ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}). \]

It is somewhat cumbersome to give an explicit description of this functor. It will therefore be more convenient to specify it implicitly by realizing it as the covariant transport representation of a left fibration over $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. We begin by discussing the counterpart of this fibration in the setting of classical category theory.

Construction 8.1.0.1 (The Twisted Arrow Category). Let $\operatorname{\mathcal{C}}$ be a category. We define a new category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as follows:

  • An object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$.

  • Let $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ be objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. A morphism from $f$ to $f'$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is a pair of morphisms $u: X' \rightarrow X$, $v: Y \rightarrow Y'$ in $\operatorname{\mathcal{C}}$ satisfying $f' = v \circ f \circ u$, so that we have a commutative diagram

    \[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^-{f} & X' \ar [l]_-{u} \ar [d]^{f'} \\ Y \ar [r]^-{v} & Y'.} \]
  • Let $f: X \rightarrow Y$, $f': X' \rightarrow Y'$, and $f'': X'' \rightarrow Y''$ be objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. If $(u,v)$ is a morphism from $f$ to $f'$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ and $(u',v')$ is a morphism from $f'$ to $f''$ in $\operatorname{\mathcal{C}}$, then the composition $(u',v') \circ (u, v)$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is the pair $(u \circ u', v' \circ v)$.

We will refer to $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as the twisted arrow category of $\operatorname{\mathcal{C}}$.

Remark 8.1.0.2. Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $(f: X \rightarrow Y) \mapsto (X,Y)$ determines a forgetful functor $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. Moreover, $\lambda $ is a left covering functor, in the sense of Definition 4.2.3.1.

Remark 8.1.0.3 ($\operatorname{Tw}(\operatorname{\mathcal{C}})$ as a Category of Elements). Let $\operatorname{\mathcal{C}}$ be a category. Then the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ determines a functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , \bullet ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$. The twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ of Construction 8.1.0.1 can be identified with the category of elements $\int _{ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet )$ (see Construction 5.2.6.1).

It follows that the functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet )$ is determined (up to canonical isomorphism) by the datum of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ together with the forgetful functor $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ of Remark 8.1.0.2 (see Corollary 5.2.7.5).

Warning 8.1.0.4 (Untwisted Arrow Categories). Let $[1] = \{ 0 < 1 \} $ denote a linearly ordered set with two elements. For any category $\operatorname{\mathcal{C}}$, we can identify morphisms of $\operatorname{\mathcal{C}}$ with functors $F: [1] \rightarrow \operatorname{\mathcal{C}}$. The collection of such functors can be organized into a category $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$, which we refer to as the arrow category of $\operatorname{\mathcal{C}}$. The arrow category $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ has the same objects as the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. However, the morphisms are different: if $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ are morphisms of $\operatorname{\mathcal{C}}$, then morphisms from $f$ to $f'$ in $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ can be identified with commutative diagrams

\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{f} \ar [r] & X' \ar [d]^{f'} \\ Y \ar [r] & Y', } \]

where the horizontal maps are oriented in the same direction.

Example 8.1.0.5. Let $Q$ be a partially ordered set, which we regard as a category. Then the twisted arrow category $\operatorname{Tw}(Q)$ can be identified (via the forgetful functor of Remark 8.1.0.2) with the partially ordered set

\[ \{ (p,q) \in Q^{\operatorname{op}} \times Q: p \leq q \} \subseteq Q^{\operatorname{op}} \times Q. \]

Remark 8.1.0.6. Let $\operatorname{\mathcal{C}}$ be a category. For every object $X \in \operatorname{\mathcal{C}}$, the fiber $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ can be identified with the coslice category $\operatorname{\mathcal{C}}_{X/}$ of Variant 4.3.1.4. Similarly, the fiber $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ X\} $ can be identified with the opposite of the slice category $\operatorname{\mathcal{C}}_{/X}$ of Construction 4.3.1.1.

In §8.1.1, we generalize Construction 8.1.0.1 to the setting of $\infty $-categories. To every simplicial set $\operatorname{\mathcal{C}}$, we associate another simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$, whose $n$-simplices can be identified with $(2n+1)$-simplices of $\operatorname{\mathcal{C}}$ (Construction 8.1.1.1). This construction has the following features:

  • If $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is (the nerve of) an ordinary category $\operatorname{\mathcal{C}}_0$, then $\operatorname{Tw}(\operatorname{\mathcal{C}})$ can be identified with the nerve of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}}_0)$ (Proposition 8.1.1.9). Consequently, the twisted arrow construction of §8.1.1 can be regarded as a generalization of Construction 8.1.0.1.

  • The simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is equipped with a projection map $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\lambda $ is a left fibration (Proposition 8.1.1.10); in particular, $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is also an $\infty $-category (Corollary 8.1.1.11).

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §8.1.2, we study the fibers of the left fibration $\lambda : \operatorname{Tw}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$. Our main result asserts that if $f: X \rightarrow Y$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$, then $f$ is initial when viewed as an object of the $\infty $-category $\{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ (see Proposition 8.1.2.1, and Corollary 8.1.2.16 for the converse). From this, we deduce an analogue of Remark 8.1.0.6: there is a canonical equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$ (Proposition 8.1.2.5), which induces a homotopy equivalence of Kan complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ X\} \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \]

for each object $Y \in \operatorname{\mathcal{C}}$ (Notation 8.1.2.10). Moreover, we show that these homotopy equivalences are compatible with covariant transport for the left fibration $\lambda $ (Corollary 8.1.2.13).

The twisted arrow construction can be characterized by a universal mapping property. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ denote the composition of $\lambda $ with the projection map $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$. Then $\lambda _{+}$ is a cocartesian fibration of $\infty $-categories (Corollary 8.1.1.12). Moreover, for each object $X \in \operatorname{\mathcal{C}}$, the fiber $\lambda _{+}^{-1} \{ X\} $ has an initial object (given by the identity morphism $\operatorname{id}_{X}$, regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$). In §8.1.3, we show that $\lambda _{+}$ is universal with respect to these properties. More precisely, we show that if $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is any cocartesian fibration having the property that each fiber $\operatorname{\mathcal{E}}_{X} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \{ X\} $ has an initial object, then there is an essentially unique functor $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ which carries each identity morphism $\operatorname{id}_{X} \in \operatorname{Tw}(\operatorname{\mathcal{C}})$ to an initial object of the $\infty $-category $\operatorname{\mathcal{E}}_{X}$. Moreover, the functor $F$ is initial when regarded as an object of the larger $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ (see Theorem 8.1.3.1 and Remark 8.1.3.2).

The twisted arrow construction $S \mapsto \operatorname{Tw}(S)$ determines a functor from the category of simplicial sets to itself. In particular, to every simplicial set $T$ we can associate a new simplicial set $\operatorname{Cospan}(T)$, whose $n$-simplices are given by maps $\operatorname{Tw}( \Delta ^ n ) \rightarrow T$. We will refer to $\operatorname{Cospan}(T)$ as the simplicial set of cospans in $T$ (Construction 8.1.4.1). This construction has the following features:

  • The construction $T \mapsto \operatorname{Cospan}(T)$ determines a functor from the category of simplicial sets to itself, which is right adjoint to the twisted arrow functor $S \mapsto \operatorname{Tw}(S)$ (Corollary 8.1.4.8).

  • Let $\operatorname{\mathcal{C}}$ be an ordinary category which admits pushouts, and let $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ denote the $2$-category of cospans in $\operatorname{\mathcal{C}}$ (Example 2.2.2.1). Then there is a canonical isomorphism of simplicial sets

    \[ \operatorname{Cospan}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) ) \xrightarrow {\sim } \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) ), \]

    which we construct in §8.1.4 (see Corollary 8.1.4.12).

  • Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $X$ and $Y$, and let $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ denote the category of $1$-morphisms from $X$ to $Y$. Then there is a canonical isomorphism of simplicial sets

    \[ \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \xrightarrow {\sim } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y), \]

    which we construct in §8.1.5 (see Corollary 8.1.5.6).

  • If $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits pushouts, then the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category (Proposition 8.1.6.1). We prove this in §8.1.6 using an explicit characterization of the collection of thin $2$-simplices of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ (Proposition 8.1.6.2), which we prove in §8.1.7.

Structure

  • Subsection 8.1.1: The Twisted Arrow Construction
  • Subsection 8.1.2: Homotopy Transport for Twisted Arrows
  • Subsection 8.1.3: The Universal Property of Twisted Arrows
  • Subsection 8.1.4: The Cospan Construction
  • Subsection 8.1.5: Morphisms in the Duskin Nerve
  • Subsection 8.1.6: Cospans in $\infty $-Categories
  • Subsection 8.1.7: Thin $2$-Simplices of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$