Restriction and addition maps

Remark 5.3.5 (Restriction and addition maps). Let \(M\colon \Span (\Fin ) \to C\) be a commutative monoid. Every map \(f\colon S \to T\) of finite sets determines two ‘half-degenerate’ spans \[ f^* \quad = \quad (T \xleftarrow {f} S \xrightarrow {\id _S} S) \qquadtext { and } f_{\oplus } \quad = \quad (S \xleftarrow {\id _S} S \xrightarrow {f} T), \] called the restriction and the addition spans, respectively. Since products in \(\Span (\Fin )\) are computed as disjoint unions by Lemma 5.3.3, the condition that \(M\) preserves finite products means that for every finite set \(S\) the map \[ (e_s^*)_{s \in S} \colon M(S) \to M^S := \prod _{s \in S} M(*) \] is an isomorphism, where \(e_s\colon \{s\} \hookrightarrow S\) is the inclusion. We will henceforth identify \(M(S)\) with \(M^S\). It follows that every map \(f\colon S \to T\) defines both a restriction map \(f^*\colon M^T \to M^S\) as well as an addition map \(f_{\oplus }\colon M^S \to M^T\), functorially in \(f\). As a special case, the maps \(\emptyset \to *\) and \(* \sqcup * \to *\) in \(\Fin \) then yield the unit and addition: \[ 0\colon * \to M \qquadtext { and } +\colon M \times M \to M. \]