Can generalize to define sheaves over an arbitrary symmetric monoidal category \(\mathsf{C}\). Define affine objects \({\mathsf{Aff}}(\mathsf{C}) \coloneqq{ {\mathsf{Comm}\mathsf{Mon}(\mathsf{C})}^{\operatorname{op}}}\) as model categories.

Relation to loop spaces and monoidal categories:

For \(\mathsf{C}\) be a symmetric monoidal category over another monoidal category \((\mathsf{D}, \otimes_D)\), and define a convolution product \begin{align*} \widehat{\otimes}: {\mathsf{Fun}}( { {\mathsf{C}}^{\operatorname{op}}}, \mathsf{D}){ {}^{ \scriptscriptstyle\times^{2} } } &\to {\mathsf{Fun}}({ {\mathsf{C}}^{\operatorname{op}}}, \mathsf{D}) \\ (F, G) &\mapsto F\widehat{\otimes} G \end{align*} where \(F\widehat{\otimes}G\) is the following left Kan extension :

- Tags: - #higher-algebra #higher-algebra/monoidal - Refs: - #todo/add-references - Links: - monoidal category