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The classic way to encounter the theory of categories is via Set Theory via the typical definition we see for categories. We see all kinds of categories that are equivalent to the category of small categories. I wonder about presentations of the theory of categories. To facilitate a discussion, we may need to define what a presentation of a theory is. It may consist of a logical language or even the standard presentations of algebraic structures. For instance, a presentation of the theory of partial monoids would count as a presentation of Categories. The presentation should come with enough structure to analyze all small Categories.
I saw Marsden put together a presentation of categories in terms of string diagrams.
I like to think that string diagrams can be seen as containers. This is a paper about containers. So the idea is that you have a (co)monad that encodes the container for the theory of categories. Could this work?
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