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The expression "up to (a) natural equivalence" is typical in mathematics and means roughly that we consider objects to be the same under certain conditions. (For example, when you look at someones face at different moments you see different pictures, but you know that the face that you observe is the same).

I've noticed, however, that mathematicians use both "up to natural equivalence" and "up to a natural equivalence". It seems to me that there is no difference in the meaning. It is true that both phrases are grammatically correct? It is possible to put this into some wider context?

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If, for every object X in C, the morphism ηX is an isomorphism in D, then η is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). (Wikipedia)

Google Books has about 600 hits each for up to a natural equivalence and up to natural equivalence, so I guess you can just use whichever you fancy. Neither is more "right" (or "wrong") than the other.

In the "mathematician's jargon" context, there's the same "optional article" in, for example, settle to (an) equilibrium. But in a more ordinary context, consider: As a blogger, I try to achieve (a) balance in my posts. Again, including the article is optional, and it doesn't affect the meaning either way.

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  • (Converted to an answer from an earlier comment)
    – J.R.
    Jun 12 '18 at 15:15

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