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The answers to this questions prove that if a category has all products and equalizers, then it has all limits. How to modify that proof to show that if a category has binary products, a terminal object, and equalizers, then it has finite limits? (This is exercise 5.1.38(b) from Leinster.)

At least, what the "set up" should be? The proof in the link heavily uses arbitrary products, and I don't see how to replace them with binary products plus terminal object. Does some "combination" of the terminal object and binary products give arbitrary products?

user557
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1 Answers1

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You do exactly the same thing, but note that the products are indexed by objects/arrows of the indexing category, so if the latter is finite, so are the products.

Then you just prove that if you have binary products and a terminal objects, you have all finite products : this is enough for the first sentence to actually go through.

Maxime Ramzi
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