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I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this theory can be helpful.

In fact, for me there is no doubt that $( \infty, 1)-$categories are really useful. For example, the work of Lurie/Toën/Vezzosi in Derived Algebraic Geometry, or the Cobordism hypothesis. I have also the idea that $( \infty, 2)-$categories are used in the new advances in Langlands.

However, I do not know any useful application of $(\infty, n)$-categories, for $n \geq 3$. I can presume that they provide the necessary formalism to fill the missing details in Lurie's proof of the Cobordism Hypothesis, but if this is so people do not seem to care too much.

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    $\begingroup$ If you know about Lurie's proof of cobordism hypothesis, then I wonder why you're so sceptical. Both the theorem and its proof are based on $(\infty,n)$-categories essentially. Personally I believe TQFTs alone would be enough of a reason to study higher categories. $\endgroup$ – Anton Fetisov Sep 30 '14 at 21:25
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    $\begingroup$ Seems to me that $(\infty, 1)$-categories are supposed to be considered as an enrichment of the concept of just "category"; i.e. they are not higher categories so much as they are deeper categories. Then $(\infty, n)$-categories are deeper, higher categories. Do you think that ordinary higher categories (i.e. 2-categories, 3-categories, etc.) are useful? If so, then their infinity version must be even more useful. $\endgroup$ – Ryan Reich Oct 1 '14 at 3:02
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In my humble opinion, the fact that such a structure appears naturally on bordisms between manifolds (eventually with some additional geometric structure like framings) is already a very good motivation to care about it.

Moreover, with such a strong relationship between fully dualizable objects in $(\infty,n)$ categories and extended TQFTs, there are strong expectations to better understand the latter (which produces invariants of manifolds) via the former and vice-versa.

I think an interesting example of such a bridge is the recent work of Douglas, Schommer-Pries and Snyder, investigating this relationship for $n=3$ to study fusion categories:

http://arxiv.org/abs/1312.7188

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  • $\begingroup$ I have heard about fusion categories and their importance to TQFTs, and I did not knew about such bridge interconnecting the latter to the theory of $(\infty,n)-$categories. Thank you for pointing out such an example! $\endgroup$ – Jorge António Oct 5 '14 at 8:54

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