Suppose we're given a strict and small 2-category $C$, and an object of $C$ called $A$. Can we produce an internal category structure on $A$ in some canonical way (maybe by some sort of argument similar to the construction of a group object [see my comment])?

If it's not possible in general, but it is possible in special (and nontrivial) situtations, what are the situations where it is possible?

If it is possible, would this be functorial (pseudofunctorial maybe?) into 2-Cat, the 2-category of small categories, and if it is, is it full, faithful, both, or neither?

If the answer to all of the above is true, does it generalize past the special case where n=2? I ask this because of the coherence result for bicategories, which fails to generalize past n=2. If this still works, can we drop the assumption of strictness?

  • $\begingroup$ Maybe the idea here would be similar to how we can construct a group object in a category by having a group structure on Hom(X,G) for all X, but now with categories. $\endgroup$ – Harry Gindi Dec 11 '09 at 1:56
  • $\begingroup$ By the way, the 2-category of categories is usually called "Cat." "2-Cat" means the (3-)category of 2-categories. $\endgroup$ – Mike Shulman Dec 11 '09 at 17:24

It's not entirely clear to me what you're asking, but I'll have a go at answering it anyway.

An internal category in a category with pullbacks consists of objects $A_0$ and $A_1$, maps $s,t\colon A_1\to A_0$, $i\colon A_0\to A_1$, and $c\colon A_1\times_{A_0}A_1\to A_1$ satisfying the usual axioms. An equivalent way to describe this is that for each object $X$ you have a category whose objects are $Hom(X,A_0)$ and whose morphisms are $Hom(X,A_1)$, which varies with $X$ in a natural way. Of course the latter makes sense even if the category has no pullbacks. This is all just like the case of groups, and it requires only an ambient category, not even a 2-category.

Now if I have a strict 2-category $C$ which has strict finite 2-limits, in particular its underlying 1-category $C^1$ has pullbacks. Moreover, any object $A$ gives rise to a canonical internal category in $C^1$ with $A_0 = A$ and $A_1 = A^{\mathbf{2}}$, the cotensor with the "walking arrow" $\mathbf{2} = (\cdot \to \cdot)$. This construction defines a strict 2-functor from $C$ to the 2-category $Cat(C^1)$ of internal categories, functors, and natural transformations in the 1-category $C^1$. Moreover, I believe that this 2-functor is strictly 2-fully-faithful, i.e. an isomorphism on hom-categories. For this reason, Street has called a strict 2-category with strict 2-pullbacks and strict cotensors with $\mathbf{2}$ (which is all you really need for this argument) a "representable 2-category": all the 2-dimensional structure of its objects can be "represented" as internal categories in some 1-category. Cf. for instance "Fibrations and Yoneda's lemma in a 2-category."

If $C$ is a non-strict 2-category, then it doesn't have an underlying 1-category, so it's not as clear how to write this down. But I think that morally, it should still be true, if one takes the care to phrase it correctly. The question of $n$-categories for $n>2$ is quite a different matter, though; I don't know if anyone's thought about it.

  • $\begingroup$ Does the strict case hold for n? It seems like the argument should work for any strict n-category satisfying those properties. $\endgroup$ – Harry Gindi Dec 11 '09 at 5:01
  • $\begingroup$ Do you mean, is there an embedding of a strict n-category C into the strict n-category of internal (n-1)-categories in C^1? I think that should probably work as long as C has strict finite n-limits. $\endgroup$ – Mike Shulman Dec 11 '09 at 17:23
  • $\begingroup$ Yeah, that's what I meant. Thanks. $\endgroup$ – Harry Gindi Dec 11 '09 at 20:25

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