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Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain complexes of finite dimensional vector spaces which takes the Euler characteristic as the decategorification map.

Is there any paper which talk about more details on this construction?

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    $\begingroup$ What kind of details? Note that it suffices to take $2$-term chain complexes; this category is equivalent to the category of Baez-Crans $2$-vector spaces (math.ucr.edu/home/baez/hda6.pdf), which as a search term should give more things to look at. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 11, 2012 at 5:52
  • $\begingroup$ Thank you for your references! By details I mean how to define sum, difference, multiplication. I think $2$-term chain complexes is enough for sum and difference but may have difficulty to define mutiplications. $\endgroup$
    – Zhaoting Wei
    Commented Sep 11, 2012 at 6:06
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    $\begingroup$ I think that this approach reflects but your own vision of this topic, so perhapes you should write a paper on it! ;-) You could also take the easier category of $\mathbb{Z}/2$-graded vector spaces. $\endgroup$
    – Fernando Muro
    Commented Sep 11, 2012 at 6:58
  • $\begingroup$ @Zhaoting: multiplication is the tensor product of chain complexes. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 11, 2012 at 7:25
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    $\begingroup$ Qiaochu: "multiplication is the tensor product of chain complexes". I presume this is exactly why the OP said that it would be difficult to define multiplication without leaving the world of 2-term complexes; the tensor product of 2-term complexes is not a 2-term complex. $\endgroup$
    – Steven Landsburg
    Commented Sep 11, 2012 at 11:36

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