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Morgan Rogers (he/him) (Sep 02 2024 at 09:03):Julius Hamilton said:
I am working through a proof that the functors form a category, with David Egolf. I am curious, if there is a deeper reason why this is so?
It's often the case that the collection of transformations into a structure inherit the components of that structure "pointwise". That is, in proving that this data forms a category, you're really only using the composition in . This can be useful, because it means when we want to define a category of "diagrams" of a particular shape in a category, we can define it as the collection of transformations into out of a structure that can have less structure than a category, such as a directed graph.
James Deikun (Sep 02 2024 at 12:32):Morgan Rogers (he/him) said:
It's often the case that the collection of transformations into a structure inherit the components of that structure "pointwise". That is, in proving that this data forms a category, you're really only using the composition in . This can be useful, because it means when we want to define a category of "diagrams" of a particular shape in a category, we can define it as the collection of transformations into out of a structure that can have less structure than a category, such as a directed graph.
I've noticed this perspective used a lot in Cisinski's book on -category theory, where diagrams shaped like arbitrary simplicial sets are often considered.