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1.4.2 The Opposite of an $\infty $-Category

Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then we can construct a new category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, called the opposite category of $\operatorname{\mathcal{C}}$, as follows:

  • The objects of the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $C,D \in \operatorname{\mathcal{C}}$, we have $\operatorname{Hom}_{ \operatorname{\mathcal{C}}^{\operatorname{op}} }( C, D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( D, C )$.

  • Composition of morphisms in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is given by the composition of morphisms in $\operatorname{\mathcal{C}}$, with the order reversed.

The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$ admits a straightforward generalization to the setting of $\infty $-categories. In fact, it can be extended to arbitrary simplicial sets.

Notation Let $\operatorname{Lin}$ denote the category whose objects are finite linearly ordered sets and whose morphisms are nondecreasing functions. Let $I$ be an object of $\operatorname{Lin}$, regarded as a set with a linear ordering $\leq _{I}$. We let $I^{\operatorname{op}}$ denote the same set with the opposite ordering, so that

\[ ( i \leq _{I^{\operatorname{op}} } j ) \Leftrightarrow (j \leq _{I} i ). \]

The construction $I \mapsto I^{\operatorname{op}}$ determines an equivalence from the category $\operatorname{Lin}$ to itself.

Recall that the simplex category $\operatorname{{\bf \Delta }}$ of Definition is the full subcategory of $\operatorname{Lin}$ spanned by objects of the form $[n] = \{ 0 < 1 < \cdots < n \} $, and is equivalent to the full subcategory of $\operatorname{Lin}$ spanned by those linearly ordered sets which are finite and nonempty (Remark There is a unique functor $\mathrm{Op}: \operatorname{{\bf \Delta }}\rightarrow \operatorname{{\bf \Delta }}$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{{\bf \Delta }}\ar [r] \ar [d]^{\mathrm{Op} } & \operatorname{Lin}\ar [d]^{I \mapsto I^{\operatorname{op}} } \\ \operatorname{{\bf \Delta }}\ar [r] & \operatorname{Lin}} \]

commutes up to isomorphism, where the horizontal maps are given by the inclusion. The functor $\mathrm{Op}$ can be described more concretely as follows:

  • For each object $[n] \in \operatorname{{\bf \Delta }}$, we have $\mathrm{Op}([n]) = [n]$ (note that the construction $i \mapsto n-i$ determines an isomorphism of $[n]$ with the opposite linear ordering $[n]^{\operatorname{op}}$).

  • For each morphism $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, the morphism $\mathrm{Op}( \alpha ): [m] \rightarrow [n]$ is given by the formula $\mathrm{Op}( \alpha )(i) = n - \alpha (m-i)$.

Construction Let $S$ be a simplicial set, which we regard as a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$. We let $S^{\operatorname{op}}$ denote the simplicial set given by the composition

\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { \mathrm{Op} } \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { S } \operatorname{Set}, \]

where $\mathrm{Op}$ is the functor described in Notation We will refer to $S^{\operatorname{op}}$ as the opposite of the simplicial set $S$.

Remark Let $S_{\bullet }$ be a simplicial set. Then the opposite simplicial set $S_{\bullet }^{\operatorname{op}}$ can be described more concretely as follows:

  • For each $n \geq 0$, we have $S_{n}^{\operatorname{op}} = S_{n}$.

  • The face and degeneracy operators of $S_{\bullet }^{\operatorname{op}}$ are given by

    \[ (d^{n}_ i: S^{\operatorname{op}}_ n \rightarrow S^{\operatorname{op}}_{n-1}) = (d^{n}_{n-i}: S_ n \rightarrow S_{n-1}) \]
    \[ (s^{n}_ i: S^{\operatorname{op}}_ n \rightarrow S^{\operatorname{op}}_{n+1}) = (s^{n}_{n-i}: S_ n \rightarrow S_{n+1}). \]

Example Let $\operatorname{\mathcal{C}}$ be a category. For each $n \geq 0$, we can identify $n$-simplices $\sigma $ of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with diagrams

\[ C_0 \xrightarrow {f_1} C_1 \xrightarrow {f_2} \cdots \xrightarrow {f_{n-1}} C_{n-1} \xrightarrow {f_ n} C_ n \]

in the category $\operatorname{\mathcal{C}}$. Then $\sigma $ determines an $n$-simplex $\sigma '$ of $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\operatorname{op}} )$, given by the diagram

\[ C_ n \xrightarrow {f_ n} C_{n-1} \xrightarrow { f_{n-1} } \cdots \xrightarrow {f_2} C_1 \xrightarrow {f_1} C_0 \]

in the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. The construction $\sigma \mapsto \sigma '$ determines an isomorphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}} \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}^{\operatorname{op}} )$.

Example Let $X$ be a topological space. Then there is a canonical isomorphism of simplicial sets $\operatorname{Sing}_{\bullet }(X) \simeq \operatorname{Sing}_{\bullet }(X)^{\operatorname{op}}$, which carries each singular $n$-simplex $\sigma : | \Delta ^{n} | \rightarrow X$ to the composite map

\[ | \Delta ^{n} | \xrightarrow {r} | \Delta ^ n | \xrightarrow {\sigma } X \]

where $r$ is denotes the homeomorphism of $| \Delta ^ n |$ with itself given by $r( t_0, t_1, \ldots , t_{n-1}, t_ n) = (t_ n, t_{n-1}, \ldots , t_1, t_0)$.

Proposition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is also an $\infty $-category.

Proof. Let $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a map of simplicial sets for $0 < i < n$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Passing to opposite simplicial sets, we are reduced to showing that the map $\sigma _0^{\operatorname{op}}: ( \Lambda ^{n}_{i})^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ can be extended to a map $( \Delta ^ n )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. This follows from our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category, since there is a unique isomorphism $( \Delta ^ n)^{\operatorname{op}} \simeq \Delta ^ n$ which carries the simplicial subset $(\Lambda ^{n}_{i})^{\operatorname{op}}$ to $\Lambda ^{n}_{n-i}$. $\square$

Remark Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will refer to the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ of Proposition as the opposite of the $\infty $-category $\operatorname{\mathcal{C}}$. Note that:

  • The objects of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ are the objects of $\operatorname{\mathcal{C}}$.

  • Given a pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the datum of a morphism from $X$ to $Y$ in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is equivalent to the datum of a morphism from $Y$ to $X$ in $\operatorname{\mathcal{C}}$.

Variant If $X$ is a Kan complex, then the opposite simplicial set $X^{\operatorname{op}}$ is also a Kan complex.