Kerodon

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

Example 2.2.2.4 (Delooping a Monoidal Category). Let $\operatorname{\mathcal{C}}$ be a monoidal category. We define a $2$-category $B\operatorname{\mathcal{C}}$ as follows:

  • The $2$-category $B\operatorname{\mathcal{C}}$ has a single object, which we will denote by $X$.

  • The category $\underline{\operatorname{Hom}}_{B\operatorname{\mathcal{C}}}(X,X)$ is the category $\operatorname{\mathcal{C}}$.

  • The composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{C}}}(X,X) \times \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{C}}}(X,X) \rightarrow \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{C}}}(X,X) \]

    is the tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.

  • The identity morphism $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{C}}}( X, X)$ is the unit object $\mathbf{1} \in \operatorname{\mathcal{C}}$.

  • The associativity and unit constraints of $B\operatorname{\mathcal{C}}$ are the associativity and unit constraints for the monoidal structure on $\operatorname{\mathcal{C}}$.

We will refer to the $2$-category $B \operatorname{\mathcal{C}}$ as the delooping of $\operatorname{\mathcal{C}}$. Note that $B\operatorname{\mathcal{C}}$ is strict as a $2$-category if and only if the monoidal structure on $\operatorname{\mathcal{C}}$ is strict (in which case we recover the delooping construction of Example 2.2.0.8). The construction $\operatorname{\mathcal{C}}\mapsto B \operatorname{\mathcal{C}}$ induces a bijection

\[ \{ \text{Monoidal Categories $\operatorname{\mathcal{C}}$} \} \xrightarrow {\sim } \{ \text{$2$-Categories $\operatorname{\mathcal{E}}$ with $\operatorname{Ob}(\operatorname{\mathcal{E}}) = \{ X \} $} \} . \]

which can be viewed as an equivalence of categories (see Remark 2.2.5.8).