Induced monoidal structure from the functor

TL;DR

This paper investigates conditions under which a monoidal structure on a subcategory can be extended to a larger category, with applications including loop spaces, providing a framework for understanding induced monoidal structures.

Contribution

It characterizes when a monoidal structure on a subcategory can be extended to the entire category, with detailed examples including loop space cases.

Findings

Conditions for extending monoidal structures are established.

Examples demonstrate the applicability to loop spaces.

Theoretical framework for induced monoidal structures is developed.

Abstract

Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$ becomes a sub monoidal category of monoidal category $\mathcal{D}$. Examples are discussed, and in particular, in an example of loop space, we elaborated all results discussed in this article.