TL;DR
This paper introduces a categorical framework for directed graphs, providing new formal tools and two descriptions of the free-properad monad, enhancing understanding of graph-based algebraic structures.
Contribution
It presents a novel categorical formalism for directed graphs with natural morphisms and subgraphs, and offers two new descriptions of the free-properad monad.
Findings
Categorical formalism for directed graphs with morphisms and subgraphs
Two descriptions of the free-properad monad—via presheaves and groupoid-enriched hypergraphs
Enhanced understanding of algebraic structures related to graphs
Abstract
A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs, second in terms of groupoid-enriched hypergraphs.
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