## Different Conceptions of Z [closed]

To the algebraist, $\mathbb{Z}$ is just the free group with one generator. To the algebraic topologist, $\mathbb{Z}$ is just the fundamental group of the circle. To be glib, what do $\mathbb{Z}$ mean to you?

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Please check Community Wiki. – Jonas Meyer Jan 22 2010 at 7:18
He he: everyone's answering the wrong question! When translated into English, the question reads "To be glib, what do the integers mean to you?". – Andrew Stacey Jan 22 2010 at 10:44
Why I don't like this question: (1) $\mathbb Z$ is not "just" anything. An "algebraist" very well might study "rings and algebras", for example. (2) It's discussion-y. How are you going to pick a "right" answer? – Theo Johnson-Freyd Jan 22 2010 at 16:29
I'm not sure how appropriate this is to MathOverflow, but I've certainly wondered the same thing and I'm interested in seeing people's answers. – Zev Chonoles Jan 22 2010 at 16:40
I agree with Theo. I don't really see what you expect to get out of asking this question. I don't like any of the answers, and it's hard to imagine somebody giving a good one. – Anton Geraschenko Jan 22 2010 at 18:36

## closed as not a real question by Pete L. Clark, Mariano Suárez-Alvarez, Theo Johnson-Freyd, S. Carnahan♦, Anton Geraschenko♦♦Jan 22 2010 at 18:36

Initial object in the category of commutative rings.

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 (Or the category of rings.) – Kevin Lin Jan 22 2010 at 7:22 (Or the category of Z-algebras) – Andrew Stacey Jan 22 2010 at 10:42

To a number theorist, shouldn't $\mathbb Z$ be the world?

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Final object in the category of schemes.

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Just $\mathbb{Z}$ ? ;-)

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An infinite discrete subset of $\mathbb R$.

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From the coarse geometry perspective, it is R! – Andrew Stacey Jan 22 2010 at 21:17