Category theory is a branch of mathematics that aims to
formalize mathematics purely in terms of "objects" and "arrows". It
can help mathematicians reason more clearly about abstract
mathematical concepts.
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A Yoneda embedding is a particular condition in category
theory.
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Index.php file is the building block/skeleton/base of a website.
Category 5 is just a category. In order to understand what category
5 means in this case, the best way would be to read up on a php, or
website building manual.