Segal Enriched Categories I

TL;DR

This paper introduces Segal M_W-categories, a generalized framework for enriched categories over a higher category with homotopy equivalences, extending classical Segal categories and including a linear version.

Contribution

It develops a new theory of Segal M_W-categories, generalizing existing notions and introducing a linear variant not previously available.

Findings

Formalism generalizes classical Segal categories

Extends enriched categories over bicategories

Provides foundational examples and future applications

Abstract

We develop a theory of enriched categories over a (higher) category M equipped with a class W of morphisms called homotopy equivalences. We call them Segal M_W -categories. Our motivation was to generalize the notion of "up-to-homotopy monoids" in a monoidal category M, introduced by Leinster. The formalism adopted generalizes the classical Segal categories and extends the theory of enriched category over a bicategory. In particular we have a linear version of Segal categories which did not exist so far. Our goal in this paper is to present the theory and provide some examples. Applications are reserved for the future.