Maclane gives a definition (in a survey on categorical algebra) of category in the following way (in my own words), a category is a two-sorted system the sorts being called the objects and the arrows and that satisfies certain properties... My questions are What is this system, is a logic, a theory? Why a category is a two-sorted system? Then, where are the categories, in another system?

bamakhosimz

bamakhosimz

Answered question

2022-09-10

Maclane gives a definition (in a survey on categorical algebra) of category in the following way (in my own words), a category is a two-sorted system the sorts being called the objects and the arrows and that satisfies certain properties...
My questions are
What is this system, is a logic, a theory? Why a category is a two-sorted system? Then, where are the categories, in another system?

Answer & Explanation

Raphael Singleton

Raphael Singleton

Beginner2022-09-11Added 19 answers

Many-sorted first-order logic is really just a notational convenience, it doesn't extend the power of first-order logic[1].
You can eliminate the multiple "sorts" using a predicate for each sort, adding axioms that everything is of some sort and nothing is of more than one sort; the other axioms of the theory have to be modified using these predicates to limit scope to things of the intended sort(s) as needed.

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