Continuous Functions on Final Comodels of Free Algebraic Theories

TL;DR

This paper generalizes Garner's algebraic theory and comodels framework for stream processors to the broader context of free algebraic theories, providing a deeper understanding of continuous functions on final comodels.

Contribution

It extends Garner's results from specific algebraic theories to the more general setting of free algebraic theories, broadening the theoretical framework.

Findings

Generalization of Garner's results to free algebraic theories

Characterization of continuous functions on final comodels

Enhanced understanding of algebraic and comodel structures in stream processing

Abstract

In 2009, Ghani, Hancock and Pattinson gave a tree-like representation of stream processors $A^{\mathbb{N}} \rightarrow B^{\mathbb{N}}$. In 2021, Garner showed that this representation can be established in terms of algebraic theory and comodels: the set of infinite streams $A^{\mathbb{N}}$ is the final comodel of the algebraic theory of $A$-valued input $\mathbb{T}_A$ and the set of stream processors $\mathit{Top}(A^{\mathbb{N}},B^{\mathbb{N}})$ can be seen as the final $\mathbb{T}_A$-$\mathbb{T}_B$-bimodel. In this paper, we generalize Garner's results to the case of free algebraic theories.