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A category (as distinguished from a metacategory) will mean any interpretation of the category axioms within set theory. Here are the details.

Sect-2,page-10, Category Theory for the Working Mathematician.

Is there an exact definition of interpretation in Mathematics? Also is there a general meta theory about interpretable structures?

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tryst with freedom
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  • You aren't interpreting "a structure" you're interpreting axioms, which is an activity in language/semantics. If you had to have a mathematical definition of it, it seems like a lot of mathematics would become circular reasoning. You may as well ask for a mathematical definition of "definition." – rschwieb Oct 04 '22 at 17:23
  • @rschwieb Actually, there is a precise mathematical definition of this concept, which is the starting point of the field of model theory within mathematical logic. When Mac Lane talks about "an interpretation of the category axioms within set theory", he means a model of the theory of categories. https://en.m.wikipedia.org/wiki/Model_theory – Alex Kruckman Oct 05 '22 at 02:20

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Per nLab,

In Categories for the Working Mathematician, Saunders Mac Lane uses the term ‘metacategory’ to mean any model of the first-order theory of categories, and reserves the word ‘category’ for a metacategory whose objects and morphisms form sets.

A category in general doesn't have to be representable by a set (a "set" depending on the foundation of mathematics you are using), in fact such categories are generally called "small". You can see in this question some examples of "large" categories.

I don't believe interpretation has any specific meaning in mathematics, it's just the word Mac Lane uses here to describe the idea of restricting the axioms only to sets.

potapeno
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