Iterated monoid and group structures on commutative monoids

Remark 5.3.15. Once we know that \(\CMon (C)\) is semiadditive (which will be proved in general in Proposition 5.3.20), the present proposition applied to \(\CMon (C)\) gives an equivalence \(\Mon (\CMon (C)) \simeq \CMon (C)\), under which the canonical monoid structure on an object \(M \in \CMon (C)\) has multiplication given by the fold map \(M \oplus M \to M\), which agrees with the algebraic addition \(+\colon M \times M \to M\) of the commutative monoid structure. As a consequence, the monoid shear map \((\pr _1,m)\) from Definition 5.1.5 coincides with the algebraic shear map \((\pr _1,+)\) from Definition 5.3.6, and there are canonical identifications \[ \CMon (\Grp (C)) \quad \simeq \quad \Grp (\CMon (C)) \quad \simeq \quad \CGrp (C). \] The first equivalence is an instance of currying functors \(\Span (\Fin ) \times \simp \catop \to C\): being a commutative monoid in the first coordinate commutes with being a group in the second. The second equivalence uses that both sides are the full subcategory of \(\Mon (\CMon (C)) \simeq \CMon (C)\) spanned by those objects whose shear map is invertible.