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I am quite new to categories and the book I am reading is Lawvere and Schanuel's Conceptual Mathematics.

At the end of Part 2 the authors use the proof of Brouwer's fixed point theorems as an example of how one can use category thinking to construct proofs, by 'objectification' and 'mapification'.

I guess this process is very important if one really want to use category to deepen understanding. However, I found this part rather difficult. I wonder whether there are some other examples that show the power of categorifying proofs.

All examples are welcome. Thanks!

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There are plenty of examples of categorification. I think that what you are referring to is exemplified nicely by the work of Baez et al. on categorifications in higher dimensional algebra. There is also the very self-contained and quite elegant work of Leinster:

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Although this is not exactly what I am looking for, it is still very interesting. Thanks! – Hui Yu Feb 20 '13 at 15:45

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