Arrow algebras

TL;DR

This paper introduces arrow algebras, a new algebraic framework that generates elementary toposes, including localic and realizability toposes, offering a versatile tool for topos construction and analysis.

Contribution

The paper defines arrow algebras and demonstrates their ability to induce various toposes, expanding the algebraic toolkit for topos theory and realizability models.

Findings

Arrow algebras induce elementary toposes via tripos-to-topos construction.

They encompass localic and realizability toposes, including those from partial combinatory algebras.

Arrow algebras have closure properties and subalgebra notions, aiding topos construction.

Abstract

In this paper we introduce arrow algebras, simple algebraic structures which induce elementary toposes through the tripos-to-topos construction. This includes localic toposes as well as various realizability toposes, in particular, those realizability toposes which are obtained from partial combinatory algebras. Since there are many examples of arrow algebras and arrow algebras have a number of closure properties, including a notion of subalgebra given by a nucleus, arrow algebras provide a flexible tool for constructing toposes; we illustrate this by providing some general tools for creating toposes for Kreisel's modified realizability.