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I was thinking on ways to define limit of a sequence (over the reals, or over a metric space, or even better, over a general topological space) using the categorical limit (final or inicial object of the image of a category by a functor). I know the caracterization of the supremum by your universal property, but I failed to find this property to the limit of a sequence.

In the most ambicious case, a general topological space, I realized that the things cannot be too simple, because a limit (categorical), when exist, is unique up to isomorphism; however, at non-Hausdorff spaces, a sequence can have a lot of limits; and moreover, these points that are limits can be such that they are not isomorphic, in the sense that we haven't an homeomorphism carrying one to the another.

So, how to translate this concept to the beautiful category language?

João Dos Reis
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    There's a really simple answer that I formulated for the case of a limit of a sequence of real numbers for Emily Riehl's category theory book. The book can be found here: http://www.math.harvard.edu/~eriehl/161/context.pdf. The example is Example 3.8.4 on page 80. – Peter Haine Jul 16 '15 at 14:46
  • It's not perfectly what I wanted, but is really good. Thanks! – João Dos Reis Jul 16 '15 at 19:09

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