Remark 4.3.3.8. Let $\operatorname{\mathcal{C}}$ be a small monoidal category. Then the presheaf category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ inherits a monoidal structure given by Day convolution (see ยง
), which is characterized up to equivalence by the following properties:
- $(1)$
The Yoneda embedding
\[ h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \bullet , C) \]can be promoted to a monoidal functor.
- $(2)$
The tensor product on $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set})$ preserves small colimits separately in each variable.
Let us specialize to the case where $\operatorname{\mathcal{C}}= \operatorname{Lin}$ is the category of finite linearly ordered sets. Note that $\operatorname{Lin}$ can be identified with a full subcategory of $\operatorname{Cat}$ which is closed under the formation of joins (and contains the unit object $\emptyset \in \operatorname{Cat}$), and therefore inherits the structure of a monoidal category (where the tensor product is given by joins). With respect to this monoidal structure, the Yoneda embedding $h: \operatorname{Lin}\rightarrow \operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ satisfies condition $(1)$ (Example 4.3.3.7), and the join functor on $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ satisfies $(2)$ by virtue of Remark 4.3.3.4. It follows that the join operation on $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$ is given by Day convolution (with respect to the join operation on the category $\operatorname{Lin}$).