On the unit of a monoidal model category

TL;DR

This paper presents a method to modify cofibrations in monoidal model categories to make the tensor unit cofibrant without changing weak equivalences, with applications to enriched categories and operads.

Contribution

It introduces a technique to adjust cofibrations ensuring the tensor unit is cofibrant, enhancing the structure of monoidal model categories.

Findings

Tensor unit becomes cofibrant after modification

Maintains original weak equivalences

Applicable to enriched categories and operads

Abstract

In this paper we show how to modify cofibrations in a monoidal model category so that the tensor unit becomes cofibrant while keeping the same weak equivalences. We obtain aplications to enriched categories and coloured operads in stable homotopy theory.