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In this contribution we investigate several extensions of the powerset thatcomprise arbitrarily nested subsets, and call them superpower set. This allowsthe definition of graphs with possibly infinitely nested nodes. additionally wedefine edges that are incident to edges. Since we use coalgebraic constructionswe refer to these graphs as corecursive graphs. The superpower set functors areexamined and then used for the definition of $\mathcal{M}$-adhesive categorieswhich are the basic categories for $\mathcal{M}$-adhesive transformationsystems. 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.

Towards M-Adhesive Categories based on Coalgebras and Comma Categories