Commutative monoids in semiadditive ∞-categories

Proof. The equivalences \(\Mon (C) \iso C\) and \(\Grp (C) \iso C\) are Proposition 5.3.14. By Exercise 5.3.12, we have \(\CGrp (C) \simeq \CMon (C) \times _{\Mon (C)} \Grp (C)\). The case for commutative groups therefore follows from that of commutative monoids, so it remains to show that the forgetful functor \(\CMon (C) \to C\) is an equivalence.

The proof strategy is similar to, but more involved than, that of Proposition 5.3.14. Consider the following subcategories of \(\Span (\Fin )\): \[ * \hookrightarrow \Fin ^{\leq 1} \hookrightarrow \Span (\Fin ^{\leq 1}) \hookrightarrow \Span (\Fin ,\inj ,\all ) \hookrightarrow \Span (\Fin ). \] Here \(\Fin ^{\leq 1} \subseteq \Fin \) is the full subcategory on sets of cardinality at most \(1\), i.e. it has objects \(\emptyset \) and \(*\) and a single non-identity morphism \(\emptyset \to *\). We claim that restriction along each of these inclusions defines equivalences \begin {align*} \CMon (C) &= \Fun ^{\times }(\Span (\Fin ),C) \iso \Fun ^{\Segal }(\Span (\Fin ,\inj ,\all ),C) \\ &\iso \Fun ^{*}(\Span (\Fin ^{\leq 1}),C) \iso \Fun ^{*}(\Fin ^{\leq 1}, C) \iso C, \end {align*}

with inverses given by right, left, right and left Kan extension, respectively. Here \(\Fun ^{*}(-,-)\) denotes those functors that send the empty set to the terminal object of \(C\). For the first equivalence, this was proved in Proposition 5.3.9. The proof strategy for the other three equivalences is similar. We will spell out the details for the second equivalence, and leave the easier third and fourth equivalences to the reader.

Denoting the relevant inclusion by \(i\colon \Span (\Fin ^{\leq 1}) \hookrightarrow \Span (\Fin ,\inj ,\all )\), we need to show that the restriction functor \[ i^*\colon \Fun ^{\Segal }(\Span (\Fin ,\inj ,\all ),C) \to \Fun ^{*}(\Span (\Fin ^{\leq 1}),C) \] is an equivalence. Note that by semiadditivity of \(C\), a functor \(F\colon \Span (\Fin ,\inj ,\all ) \to C\) satisfies the Segal condition if and only if its restriction \(F\vert _{\Fin }\colon \Fin \to C\) is left Kan extended from the point. Similarly, a functor \(G\colon \Span (\Fin ^{\leq 1}) \to C\) satisfies \(G(\emptyset ) = *\) if and only if its restriction \(G\vert _{\Fin ^{\leq 1}}\colon \Fin ^{\leq 1} \to C\) is left Kan extended from the point. We now claim that left Kan extension along \(i\) restricts to a functor \[ i_!\colon \Fun ^{*}(\Span (\Fin ^{\leq 1}),C) \to \Fun ^{\Segal }(\Span (\Fin ,\inj ,\all ),C). \] To see this, let \(G\colon \Span (\Fin ^{\leq 1}) \to C\) be a functor satisfying \(G(\emptyset ) = *\). It will suffice to show that the restriction \(i_!(G)\vert _{\Fin }\) of the left Kan extension of \(G\) is itself the left Kan extension of \(G\vert _{\Fin ^{\leq 1}}\), since then it must be Kan extended from the point. If we let \(i'\colon \Fin ^{\leq 1} \to \Fin \) denote the inclusion, there is a canonical map \[ i'_!(G\vert _{\Fin ^{\leq 1}}) \to i_!(G)\vert _{\Fin }. \] Using the pointwise formula for Kan extensions, Theorem 21.4.3, we may compute both sides as colimits of \(G\) over the relative slices of \(i'\) and \(i\), and the previous comparison map is induced by the inclusion map \(\Fin ^{\leq 1}_{/S} \to \Span (\Fin ^{\leq 1})_{/S}\) of relative slices, for \(S \in \Fin \). Since this inclusion admits a left adjoint by Lemma 5.3.16, this functor is final by Corollary 21.5.7, and hence restriction along it does not change the value of the colimit.

Finally we observe that the Kan extension \(i_!G\) has the same value on the point as \(G\) itself. From this one concludes that the unit \(G \to i^*i_!G\) and counit \(i_!i^*F \to F\) of the adjunction are both natural isomorphisms, and hence these two functors are inverse equivalences. □