Lemme de Yoneda pour les foncteurs \`a valeurs monoidales

TL;DR

This paper extends the Yoneda Lemma to monoidal valued functors within a closed symmetric monoidal category, establishing that functor categories inherit a closed symmetric monoidal structure and deriving related adjoint functor theorems.

Contribution

It generalizes the Yoneda Lemma for monoidal valued functors and demonstrates that functor categories are closed symmetric monoidal categories.

Findings

$ ext{Hom}$-functors are representable in this setting

Functor categories inherit a closed symmetric monoidal structure

Derived an adjoint functor theorem for monoidal valued functors

Abstract

We consider a closed symmetric monoidal category $\mathcal{M}$. We show that if $I$ is a small category then $\mathcal{M}^I$ is a closed $\mathcal{M}$-module. We rewrite the Yoneda Lemma in the case of monoidal valued functors. We derive an adjoint functor theorem and we show that $\mathcal{M}^I$ is a closed symmetric monoidal category