Segal Enriched Categories I

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.

💡 Research Summary

The paper introduces a new framework called “Segal M_W‑category,” which generalizes both classical Segal categories and enriched category theory by allowing a higher category M equipped with a distinguished class W of morphisms—called homotopy equivalences—to serve as the enriching base. The authors begin by fixing a higher category M (which may be a 1‑category, a bicategory, or even a linear category such as chain complexes) together with a subcollection W of its 1‑cells (or 2‑cells) that satisfies the usual closure properties (contains identities, closed under composition, and satisfies a 2‑out‑of‑3 condition). This class plays the role of “weak equivalences” and is used to relax strict associativity and unit laws.

A Segal M_W‑category consists of a set of objects X and a simplicial object C· in the category of M‑enriched diagrams. For each n≥0, C_n is an object of M, and the face and degeneracy maps are M‑morphisms. The central Segal condition is reformulated in the W‑local setting: for every n≥2 the canonical map
C_n → C_1 ×{C_0} … ×{C_0} C_1
must belong to W. In other words, the n‑ary composition diagram is required to be a W‑equivalence rather than an isomorphism. This yields a “up‑to‑homotopy” composition law that is coherent in the sense of higher category theory.

The authors verify that the definition recovers known structures in several special cases. When M is the ordinary category of sets and W consists of all bijections, a Segal M_W‑category is exactly a classical Segal category. When M is a monoidal category with a single object and W is the class of monoidal weak equivalences, the notion coincides with Leinster’s “up‑to‑homotopy monoid.” When M is a bicategory, the construction reproduces the established theory of categories enriched over a bicategory. Thus the new definition truly unifies these previously disparate frameworks.

A particularly novel contribution is the introduction of a linear version of Segal categories. Here M is taken to be the differential graded (DG) category of chain complexes over a field k, and W is the class of quasi‑isomorphisms. In this setting a Segal M_W‑category provides a homotopy‑coherent, DG‑enriched category where composition is defined only up to quasi‑isomorphism. Prior to this work no linear analogue of Segal categories existed, and the authors demonstrate how this structure naturally encodes derived tensor products and higher homotopies in homological algebra.

The paper also establishes basic categorical properties of Segal M_W‑categories. After localizing M at W, the resulting homotopy category M