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Let $\mathcal{C}$ be a $1$-category. One can show that the category of internal categories in $\mathcal{C}$, with internal functors and internal natural transformations produce a $2$-category.

I find this rather surprising, considering that an internal object in $\mathcal{C}$ is a functor (generally continuous and stuff) from a diagram category to $\mathcal{C}$, while internal morphisms are natural transformations between such functors. Thus, there naturally exists a structure of 1-category associated to internal objects, but a priori no 2-categorical structure. I tried to check if the concept of internal transformation could be encoded as a sort of "modification" using 1-cells only, but it seems to fail hard. This means that the 2-cells structure is "non canonical" and has to be put by hand. Hence my question: is there a maximal $n$ such that internal objects in a $1$-catetgory with their internal cells up to $n$ forms an $n$-category?

PS: I guess I would expect such $n$ to be 2 because the only "internal structure" of an object is given by its generalized elements (that is, arrows), but I feel confused now.

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Technically there is no maximum. You could create ∞-categories from a 1-category, but anything higher than the initial 1-category would be trivial.

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  • $\begingroup$ I don't think you read the question. It's not asking about how to turn a 1-category into an n-category, but rather asking about internal categories in a 1-categories and so on. $\endgroup$ – Najib Idrissi Oct 5 '15 at 7:38
  • $\begingroup$ @user: that's not even true. Any 1-category can be turned into a double category canonically (same objects, same vertical and horizontal arrows, and commutative squares as 2-cells). You can even iterate this process with cubes, and so on. This is not disturbing here, because such higher structure can easily be seen as a (double)-functor (double)-category in a canonical way (hence, it is sketchable). What disturbs me in my question is that the higher category structure of internal categories of C not visible as a (2)-functor (2)-category, and is a priori not sketchable. How comes ? $\endgroup$ – sure Oct 5 '15 at 8:38

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