# Reflecting Algebraically Compact Functors

## TL;DR

This paper introduces a new abstract method for constructing algebraically compact functors in categories lacking zero objects or limit-colimit coincidences, expanding the theoretical framework for recursive datatype semantics.

## Contribution

It provides a constructive approach to building compact algebras in more general categories by reflecting from categories with known properties, improving upon existing methods.

## Key findings

- Constructs compact algebras without zero objects or limit-colimit coincidences.
- Provides a constructive description of a broad class of algebraically compact functors.
- Demonstrates the advantages of the new approach over previous methods.

## Abstract

A compact T-algebra is an initial T-algebra whose inverse is a final T-coalgebra. Functors with this property are said to be algebraically compact. This is a very strong property used in programming semantics which allows one to interpret recursive datatypes involving mixed-variance functors, such as function space. The construction of compact algebras is usually done in categories with a zero object where some form of a limit-colimit coincidence exists. In this paper we consider a more abstract approach and show how one can construct compact algebras in categories which have neither a zero object, nor a (standard) limit-colimit coincidence by reflecting the compact algebras from categories which have both. In doing so, we provide a constructive description of a large class of algebraically compact functors (satisfying a compositionality principle) and show our methods compare quite favorably to other approaches from the literature.