Enriched weakness

TL;DR

This paper introduces enriched weak notions in category theory, replacing strict uniqueness with a surjection condition on hom-objects, and explores their role in describing homotopy coherent structures.

Contribution

It develops enriched versions of weak categorical notions using surjections, extending the framework of category theory to include homotopy coherence.

Findings

Enriched weak notions are characterized by morphisms belonging to a class of surjections.

The framework applies to describe homotopy coherent structures.

Provides a new perspective on injectivity and weak orthogonality in enriched categories.

Abstract

The basic notions of category theory, such as limit, adjunction, and orthogonality, all involve assertions of the existence and uniqueness of certain arrows. Weak notions arise when one drops the uniqueness requirement and asks only for existence. The enriched versions of the usual notions involve certain morphisms between hom-objects being invertible; here we introduce enriched versions of the weak notions by asking that the morphisms between hom-objects belong to a chosen class of "surjections". We study in particular injectivity (weak orthogonality) in the enriched context, and illustrate how it can be used to describe homotopy coherent structures.