# Compactness in abelian categories

## TL;DR

This paper generalizes the concept of compact objects in abelian categories relative to a subclass, demonstrating that many closure properties hold and identifying conditions for products of compact objects to remain compact.

## Contribution

It introduces a relativized notion of compactness in abelian categories and explores conditions ensuring the stability of compactness under products.

## Key findings

- Closure properties of compact objects are preserved in the relativized setting.
- Conditions are identified under which products of compact objects remain compact.
- The paper extends classical results to a broader categorical context.

## Abstract

We relativize the notion of a compact object in an abelian category with respect to a fixed subclass of objects. We show that the standard closure properties persist to hold in this case. Furthermore, we describe categorical and set-theoretical conditions under which all products of compact objects remain compact.