Commutative monoids via the Segal condition

Proof. It is clear from the definitions that restriction along \(i\) sends product-preserving functors to functors satisfying the Segal condition.

The functor \(i^*\) is conservative: if \(\alpha \colon M \to N\) is a natural transformation between commutative monoids such that \(i^*\alpha \) is an isomorphism, then in particular \(\alpha _*\colon M(*) \to N(*)\) is an isomorphism, and since both \(M\) and \(N\) preserve products we have that the map \(\alpha _S\colon M(S) \cong M(*)^S \to N(*)^S \cong N(S)\) is an isomorphism for all finite sets \(S\).

It thus suffices to show that for every functor \(M\colon \Fin _* \to C\) satisfying the Segal condition, the right Kan extension \(i_*(M)\) exists and the counit map \(i^* i_*(M) \to M\) is an isomorphism. By the pointwise formula for right Kan extensions, we must show that for every finite set \(X\) the canonical map \[ \lim _{(X \to S) \in i_{X/}} M(S) \longrightarrow M(X) \] is an isomorphism, where the limit is taken over the relative slice category \(i_{X/} = \Span (\Fin ,\inj ,\all ) \times _{\Span (\Fin )} \Span (\Fin )_{X/}\). Consider the embedding \[ \phi \colon (\Fin \catop )_{X/} \times _{\Fin \catop } \Fin _{\inj }\catop \hookrightarrow \Span (\Fin ,\inj ,\all ) \times _{\Span (\Fin )} \Span (\Fin )_{X/} \] that sends a map \(f\colon S \to X\) to the span \(X \xleftarrow {f} S \xrightarrow {\id } S\). This functor \(\phi \) has a right adjoint, given by sending a span \(X \leftarrow U \to S\) to the left-pointing map \(U \to X\). By Corollary 21.5.7, the functor \(\phi \) is therefore initial, so the limit may be computed over the left-hand category.

On this category, the restriction of \(M\) sends a map \(f\colon S \to X\) to \(M(S) \cong M(*)^S = \lim _{s \in S} M(*)\) (using the Segal condition). By the formula for iterated limits from Theorem 21.2.11, we may thus write the left-hand side as the limit of \(M(*)\) over the set \(\colim _{(X \to S)} S\). It thus suffices to show that \(X\) is the colimit of the forgetful functor \[ \Fin _{/X} \times _{\Fin } \Fin _{\inj } \longrightarrow \Fin . \] This functor is left Kan extended from the full subcategory spanned by the inclusions \(\{x\} \hookrightarrow X\) for \(x \in X\), and the colimit over this subcategory is precisely \(X\). Moreover, the product projections in \(\Span (\Fin )\) have injective left legs and therefore belong to the wide subcategory on which \(i\) is defined. The product comparison maps for \(i_*M\) consequently agree, under the counit, with the Segal maps of \(M\) and are isomorphisms. Thus \(i_*M\) preserves finite products. □