# Kerodon

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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.