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I'm just beginning my journey in learning category theory and its generality is difficult to grapple with. I'm trying to find mental tricks to tie it down a bit.

From my current naive position, it seems like there are two flavors of categories: structural and operational, i.e. categories that define a static structure versus categories that define a dynamic process. Is this reasonable to say or are there categories for which neither concept makes sense to apply?

For example, categories like poset seem purely structural, where objects are static things and morphisms are just relations (e.g. $\leq$ ). On the other hand there are categories like monoidal categories where it is natural to think of them as processes, hence we can use string diagrams.

Or since categories are basically just (multi)graphs with enforced composition of edges, I think of graphs like a social network as static data structures vs a flow chart that is a process.

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    $\begingroup$ I mean, why not both? $\endgroup$
    – Randall
    Commented Jan 10, 2018 at 21:47
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    $\begingroup$ What is a benefit of labelling categories "structural" or "operational"? Are there useful theorems that can involve this notion and be proven? I can see how it is useful for you, from the didactical point of view, but I don't think this division has a wider significance. It can also obscure the fact that what you call "structural" and what you call "operational" categories can be treated equally using the machinery of the category theory, in effect completely erasing the distinction. $\endgroup$
    – user491874
    Commented Jan 10, 2018 at 21:52
  • $\begingroup$ @Randall Can you point to a category that is both? And if there are categories that are both structural and operational, wouldn't it still be possible to distinguish the structural aspects and process aspects? $\endgroup$ Commented Jan 10, 2018 at 21:53
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    $\begingroup$ @user8734617 You're right, there is no benefit in terms of using category theory, it is purely a semantic difference that is of didactic use (if true). Thinking like this is just training wheels for me. I imagine that as I get more comfortable, I won't need to employ such a distinction. $\endgroup$ Commented Jan 10, 2018 at 21:55
  • $\begingroup$ Honestly, no, because I cannot see them that way. $\endgroup$
    – Randall
    Commented Jan 10, 2018 at 21:56

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