You may be interested in my arXiv paper: Causal-net category, https://arxiv.org/abs/2201.08963.

I totally agree with
Mirco A. Mannucci \bib{384818}{misc}{

title={Graph theory from a category theory perspective},

author={Mirco A. Mannucci (https://mathoverflow.net/users/15293/mirco-a-mannucci)},

note={URL: https://mathoverflow.net/q/384818 (version: 2021-02-24)},

eprint={https://mathoverflow.net/q/384818},

organization={MathOverflow}

}

The main idea to view graph theory from the category theory perspective is summarized by the slogan:
Graph Theory= Kleisli Category.

In this paper, we introduce the category of causal-nets, called causal-net category and denoted by Cau, which takes causal-nets as objects and the functors between the path categories of causal-nets as morphisms. This category is actually the Kleisli category of the “free category on a causal-net” monad.

The introduction of this category is motivated by the study of causal-net condensation, which aims to give a categorical formulation of the Baez construction of spin networks for gauge theory. Causal-net category plays the role in causal-net condensation just as that of little n-disc operad in factorization homology. The construction of causal-net condensation turns out essentially to be a reformation of Joyal-Street’s graphical calculus for symmetric monoidal categories. Through detailed analysis, we found that there are six types of fundamental morphisms in Cau, which correspond exactly to six basic conventions of graphical calculi for monoidal categories. The correspondence between fundamental morphisms and basic conventions shows a fundamental connection between graph theory and monoidal categories.

In this paper, we study several composition-closed classes of morphisms, such as coarse-grainings, mergings, contractions, fusions, immersions, etc. We show that many common graph operations can be represented by morphisms of Cau in the same direction or in the opposite direction. We also introduce a purely categorical framework for general minors, which is meaningful for general small categories. We categorify the usual minor posets to gauged minor categories, which turns out to be acyclic categories. Based on this results, we conclude that causal-net category is a natural framework for the study of causal-nets and the theory provides a new understand of categorical graph theory.

In this paper, we only discuss acyclic directed graphs, but the way of discussion is effectively suitable for general directed graphs and undirected graphs.

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