Abelian groups

Example 16.4.4 (Abelian groups). Take \(C=\Set \) with its cartesian monoidal structure. Then \(\CGrp (\Set )=\Ab \), and the symmetric monoidal structure supplied by Proposition 16.4.1 is the usual tensor product of abelian groups. Indeed, regard abelian groups \(A\), \(B\) and \(C\) as product-preserving functors \(\Span (\Fin )\to \Set \), so that \(A(S)\cong A^S\), and similarly for \(B\) and \(C\). By the universal property of Day convolution and the fact that \(C\) is local, maps from the localized Day convolution of \(A\) and \(B\) to \(C\) correspond to families of maps \[ A(S)\times B(T)\longrightarrow C(S\times T) \] that are functorial in the finite sets \(S\) and \(T\) and their spans.

Evaluating such a family at \(S=T=*\) gives a map \(\beta \colon A\times B\to C\). Compatibility with the spans encoding addition shows that \(\beta \) is additive in each variable. Conversely, a bilinear map \(\beta \colon A\times B\to C\) determines the family \[ \bigl ((a_s)_{s\in S},(b_t)_{t\in T}\bigr ) \longmapsto \bigl (\beta (a_s,b_t)\bigr )_{(s,t)\in S\times T}, \] and bilinearity is exactly what is needed for compatibility with composition of spans. We therefore obtain natural bijections \[ \Hom _{\CGrp (\Set )}(A\otimes B,C) \cong \{\text {Bilinear maps $A\times B\to C$}\} \cong \Hom _{\Ab }(A\otimes _{\Z }B,C). \] By Yoneda, the tensor product furnished by the Day convolution localization is the usual tensor product \(A\otimes _{\Z }B\).