Flabby and injective objects in toposes

TL;DR

This paper introduces a general concept of flabby objects in elementary toposes, explores their properties, and demonstrates their applications in understanding injective objects, cohomology, and flabby sheaves across different topos contexts.

Contribution

It defines flabby objects in elementary toposes, links internal and external injective objects, and applies these ideas to higher direct images and the effective topos.

Findings

Flabby objects unify notions across topos types.

Internal and external injective objects coincide in this setting.

Higher direct images can be interpreted as internal cohomology.

Abstract

We introduce a general notion of flabby objects in elementary toposes and study their basic properties. In the special case of localic toposes, this notion reduces to the common notion of flabby sheaves, yielding a site-independent characterization of flabby sheaves. Continuing a line of research started by Roswitha Harting, we use flabby objects to show that an internal notion of injective objects coincides with the corresponding external notion, in stark contrast with the situation for projective objects. We show as an application that higher direct images can be understood as internal cohomology, and we study flabby objects in the effective topos.