Example 2.2.0.8 (00ES): Delooping

Example 2.2.0.8 (Delooping). Let $\operatorname{\mathcal{M}}$ be a category equipped with a strict monoidal structure $\otimes : \operatorname{\mathcal{M}}\times \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}$ (Definition 2.1.2.1). We define a strict $2$-category $B\operatorname{\mathcal{M}}$ as follows:

The set of objects $\operatorname{Ob}( B\operatorname{\mathcal{M}})$ is the singleton set $\{ X \} $.
The category $\underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X )$ is equal to $\operatorname{\mathcal{M}}$.
The composition functor $\circ : \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X ) \times \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X ) \rightarrow \underline{\operatorname{Hom}}_{B\operatorname{\mathcal{M}}}( X, X )$ is equal to the tensor product $\otimes : \operatorname{\mathcal{M}}\times \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}$.
The identity morphism $\operatorname{id}_{X}$ is the strict unit object of $\operatorname{\mathcal{M}}$.

We will refer to $B\operatorname{\mathcal{M}}$ as the delooping of $\operatorname{\mathcal{M}}$.

Note that the constructions

\[ \operatorname{\mathcal{M}}\mapsto B\operatorname{\mathcal{M}}\quad \quad \operatorname{\mathcal{C}}\mapsto \underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X) \]

induce mutually inverse bijections

\[ \{ \text{Strict Monoidal Categories $\operatorname{\mathcal{M}}$} \} \simeq \{ \text{Strict $2$-Categories $\operatorname{\mathcal{C}}$ with $\operatorname{Ob}(\operatorname{\mathcal{C}}) = \{ X \} $} \} , \]

generalizing the identification of Remark 1.3.2.4.