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in this post on math.stackexchange, the top voted answer affirmed the following quote:

Roughly speaking, category theory is graph theory with additional structure to represent composition.

I am wondering, are networks categories? s and if so, is category theory ever considered when studying the functioning of various kinds of multi-layer dynamic networks, be they social, biological, etc?

neutrino
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    No; as the quoted text says, you need "additional structure to represent composition" and networks / graphs don't have this structure. – Qiaochu Yuan Aug 29 '22 at 21:24
  • could we build that additional structure into network models in order to study things that have been traditionally modeled by networks? – neutrino Aug 29 '22 at 23:20
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    You can do anything you want, but as far as things that other people have done, there seems to be more interest in thinking, more or less, about the category of networks than about networks as categories in themselves, see eg https://arxiv.org/abs/2006.10733. This makes sense, since connections in networks are often simply not composable/transitive: the friend of my friend is not in general my friend, and we want to keep track of the difference! – Kevin Carlson Aug 30 '22 at 01:06
  • A graph is certainly a tempting mental image to use in conceptualizing a category, but order to appreciate the necessary structure lacking from a graph, it is helpful to consider graphs with multiple edges between nodes -- the starkest case is of a single node with multiple edges. Given two such self-loops, how is one to compose them? – PrimeRibeyeDeal Aug 30 '22 at 02:53
  • @KevinArlin how about organizational colleagues for example? i might not be connected to my coworkers coworker directly in terms of projects, but they are my coworker. – neutrino Aug 30 '22 at 05:28
  • @PrimeRibeyeDeal can you elaborate? – neutrino Aug 30 '22 at 05:30
  • @neutrino If a graph has a single node, then for it to have a composition law, any two edges must compose to some edge on the same node. A category with one object is equivalent to a (associative) monoid, and so that's the additional structure your one-node graph lacks. And here I'm only thinking directed graphs. If they're undirected you have yet more structure to add. – PrimeRibeyeDeal Aug 30 '22 at 12:18
  • @PrimeRibeyeDeal Actually a category with one object is not equivalent to a monoid. A pointed category with one object is. See https://math.stackexchange.com/questions/4519520/why-conceptually-isnt-the-2-category-of-connected-groupoids-equivalent-to-the – Carla only proves trivial prop Aug 30 '22 at 13:26
  • @Carlaonlyprovestrivialprop What is the difference between a one-object category and a pointed one-object category? – PrimeRibeyeDeal Aug 30 '22 at 16:15
  • @PrimeRibeyeDeal It's just in the natural transformations between functors between them, not really relevant to this discussion. – Kevin Carlson Aug 30 '22 at 16:57
  • @neutrino That example is vague. There is a graph where edges between people indicate that those people work at the same company. There is another graph where such edges indicate two people who work on the same project. These are related, but if you forcibly add a composition law to the second graph, which is called taking the transitive closure of the corresponding relation, you probably still won't have recovered the first graph. – Kevin Carlson Aug 30 '22 at 17:23

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