Bases as Coalgebras

TL;DR

This paper explores coalgebras arising from the free algebra adjunction, demonstrating how they serve as bases for elements in various algebraic structures, including vector spaces and frames, and relate to comonoid structures.

Contribution

It introduces a novel perspective on bases as coalgebras of a comonad induced by the free algebra adjunction, extending to structures like vector spaces and frames.

Findings

Coalgebras can be understood as bases decomposing elements into primitives.

These coalgebras induce comonoid structures for copy and delete operations.

Application to structures like vector spaces, dcpos, and frames.

Abstract

The free algebra adjunction, between the category of algebras of a monad and the underlying category, induces a comonad on the category of algebras. The coalgebras of this comonad are the topic of study in this paper (following earlier work). It is illustrated how such coalgebras-on-algebras can be understood as bases, decomposing each element x into primitives elements from which x can be reconstructed via the operations of the algebra. This holds in particular for the free vector space monad, but also for other monads, like powerset or distribution. For instance, continuous dcpos or stably continuous frames, where each element is the join of the elements way below it, can be described as such coalgebras. Further, it is shown how these coalgebras-on-algebras give rise to a comonoid structure for copy and delete, and thus to diagonalisation of endomaps like in linear algebra.