In this paper we develop a theory of Segal enriched categories. Our motivation was to generalize the notion of up-to-homotopy monoid in a monoidal category, introduced by Leinster. Our formalism generalizes the classical theory of Segal categories and extends the theory of categories enriched over a 2-category. We introduce Segal dg-categories which did not exist so far. We show that the homotopy transfer problem for algebras leads directly to a Leinster-Segal algebra.
Keywords: Segal categories, enriched categories, homotopy algebras, higher categories
2020 MSC: 18D20, 18G30, 18N10, 18N40, 18N45, 18N60
Theory and Applications of Categories, Vol. 35, 2020, No. 33, pp 1227-1267.