Internal Category explained

Question

Would someone be able to provide me with an intuitive explanation of what an internal category is? In the usual definition we have an object of objects \(X_0 \in K\)and an object of morphisms \(X_1 \in K\) with some commutative diagrams being satisfied. \(K \)is the category inside which we defined the internal category.

The proper definition can be found here http://ncatlab.org/nlab/show/internal+category . 

Thanks a lot.

Comments

The link has two ``http://''s; that should be changed.

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You might want to examine what an "internal group" or "internal monoid" is, first, just to get an idea of what internalization looks like.

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Thanks a lot for your comment. Thanks to Urs' help I have some control over the subject now. 

Answer 1

For applications in physics, maybe best to think of the ambient category as being one of certain geometric spaces.

For instance if the ambient category is that of smooth manifolds, then a category internal to that consists of a smooth manifold of objects and a smooth manifold of morphisms, such that all operations (source, target, identity, composition) are smooth functions between these.

This is well known in the case that the category happens to be a groupoid (the case that all morphisms are invertible). A groupoid internal to the category of smooth manifolds is just a Lie groupoid, the groupoid-generalization of a Lie group.

In physics the best well known example of this is maybe the concept of an orbifold: an orbifold is a Lie groupoid, hence an internal groupoid, hence an internal category (internal in smooth manifolds). This really just means that on the collection of all the points and that of all orbifold transformations between them ("twisted sectors") there is suitable smooth structure, and composition of orbifold transformations etc. is a smooth operation. For exposition of how this works in detail see Moerdijk 02.

If you have other examples of internalization contexts in mind for which you would like to gain more intuition, let me know what your context is and what motivates you, and I'll further expand in that direction.

Comments

Hi and thanks for the answer. I would like just a more "wordy" explanation of what an internal category is. Categories contain objects and morphisms and one uses functors to relate categories. What is the role of an internal category within a category K? I just have it mixed up in my mind. This also is buffling me. I do not want any physics related explanation. 

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Let's see. So maybe first of all it is good to know that also plain categories are internal, namely internal to the category of sets. Think of the big category of sets as the standard backdrop in which mathematics takes place. Saying that any other category is a set of objects with a set of morphisms and functions of sets between these exhibiting composition etc. is really saying that any category is a category internal to sets. This just means that the stuff it's made of are sets. 

Now one grand idea in category theory is that any other big category may take the place of the category of sets and be regarded as an alternative backdrop in which to do mathematics. This works best if the big category has some nice properties which make one call it a "topos", but for our purposes here it may be good to ignore this for the moment.

So for instance there is the category of smooth manifolds that I mentioned in my reply above. You may think of this as being like the category of sets, only that the sets we consider now carry extra geometric structure. They are not just a bunch of points, but are points that hang together smoothly (they have "cohesion" among them). Now we may take *that* as the ambient mathematical universe and try to do all our mathematics inside the category of such "smooth sets". This will have the result that wherever we previously spoke of sets, we will now speak of smooth sets, and wherever we previously talked about functions between sets, we will now talk about smooth functions. So now a category internal to that mathematical universe is a "smooth category" in the same sense that a Lie group is a "smooth group".

As I said, this perspective of regarding the big ambient category as an "ambient mathematical universe" works best if the ambient category is nice enough. For that one should really consider the ambient category of smooth spaces instead of just that of smooth manifolds. But, as I said, maybe for the moment best to ignore this point.

In fact, and maybe this is what you  are after to get intuition for, one may in principle also take very puny categories and regard them as "ambient mathematical universes" in which to do everything else. For instance once could take the category of finite dimensional vector spaces as the ambient "universe" and ask what a category internal to that would be. So that's now a vector space of objects and a vector space of morphisms with certain linear maps between them. Again, you are invited to think of the vector spaces here as replacing plain sets, in that these are now sets equipped with "linear structure". A category internal to vector spaces turns out to be like a chain complex with two entries, a "2-vector space".

And so forth. But I'll stop here and wait for your reaction and further questions first.

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Thank you so much for your answer. The reason I have delayed commenting is because I have been trying to put my thoughts in order. 

Could you please explain in further detail the part of the 2-vector space and how I can consider it as an internal category? It is an example you give and helps a lot. I would really appreciate it.

Thanks a lot.

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Check out the pedagogical discussion in sections 2 and 3 of arXiv:0307263

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Thank you very much!