Towards M-Adhesive Categories based on Coalgebras and Comma Categories

TL;DR

This paper introduces superpower sets for defining infinitely nested graphs with edges incident to edges, and explores their coalgebraic structures to establish M-adhesive categories for transformation systems.

Contribution

It extends powerset concepts to superpower sets, enabling the modeling of infinitely nested graphs and establishing conditions for M-adhesive categories using coalgebraic constructions.

Findings

Superpower sets allow nested graphs with edges incident to edges.

Coalgebraic categories are M-adhesive if functors preserve pullbacks along monomorphisms.

Coalgebras in Sets are M-adhesive under certain functor conditions.

Abstract

In this contribution we investigate several extensions of the powerset that comprise arbitrarily nested subsets, and call them superpower set. This allows the definition of graphs with possibly infinitely nested nodes. additionally we define edges that are incident to edges. Since we use coalgebraic constructions we refer to these graphs as corecursive graphs. The superpower set functors are examined and then used for the definition of $\mathcal{M}$-adhesive categories which are the basic categories for $\mathcal{M}$-adhesive transformation systems. So, we additionally show that coalgebras $\mathbf{Sets}_F$ are $\mathcal{M}$-adhesive categories provided the functor $F:\mathbf{Sets}_F \to \mathbf{Sets}_F$ preserves pullbacks along monomorphisms.