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Still wading through higher category theory. I find the subject a bit intimidating, not so much for technical reasons, but because I lack sufficient intuition as to the motivation(s)/heuristics one should use a particular definition instead of another one.

The plethora of formal possibilities is so great that I would love to have a road map of sorts (such as: if you want to do this, follow this choice, if you wanna do that, here is the menu)

I do know that there are motivations, for instance in the context of abstract homotopy theory, abstract quantum field theory, etc.

But I wonder:

from a DATA MODELING's standpoint, is there any research geared toward using higher cats as advanced data structures?

After all, graphs and ordinary 1-dim cats are extremely useful in this respect, so it seems to me that their higher version should also play a big role.

Any good refs, thoughts?

CODA: The ideal situation I have in mind would be something like a nice handbook, titled

--- higher categories for the working computer scientist--

or

---higher categories for the working data modeler---.

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3 Answers

There is the recent applied category theory work of David Spivak: http://math.mit.edu/~dspivak/informatics/

Particularly, his work on simplicial databases may be of interest to you.

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Thanks Jeremy, yes, it does seem to be relevant –  Mirco Mannucci Jul 15 '12 at 20:43
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There is research which you can find by a web search on "higher categories and concurrency". The basic intuition is that handling many computers for example accessing databases you can regard each as having it's own time, so we are dealing with $n$-dimensional time.

You can also search on "double categories and data structures".

My own research has been in using (strict) double and higher groupoids in homotopy theory. This gets round the old problem that double groups are abelian groups; double groupoids are surprisingly complicated, and are in a sense "more nonabelian" than groups.

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@Ronnie Thank you so much! The connection with multiple databases (which I ignored) is especially appealing to me, as one problem I do have in my "real" life is this: there is now an entire new zoo of noSQL databases (for instance graph databases, document-based databases, etc) and it occurred to me that each has its own merit and strength, but none solves all the problems. So the issue is: use several different types of DBs, and let them work separately but in a concerted way. Perhaps higher cats could be a good weapon for this job... –  Mirco Mannucci Jul 15 '12 at 18:05
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There is work on higher categories and rewriting theory. This considers higher categories as a way of encoding the resolutions of a theory which may itself be of a class of categories. The relevant names are Burroni, Metayer, Lafont, Malbos, Guiraud, so look them up and checkout their preprints etc.

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@Tim thanks! That also looks like something I need to delve into. Any survey paper you know of? –  Mirco Mannucci Jul 16 '12 at 17:44
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It depends on exactly which bit you need. You could start with the papers and preprints on Yves Guiraud's page [pps.univ-paris-diderot.fr/~guiraud]. –  Tim Porter Jul 17 '12 at 5:27
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