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A category can be regarded as a set with certain algebraic operations. Hence we can define a category object in a category just like a group object. Is this notion useful? Is there any research on them?

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Are you asking about a category of categories? –  Asaf Karagila Jan 7 '13 at 23:58
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@AsafKaragila Not at all. Please read the link to understand what I am talking about. –  Makoto Kato Jan 8 '13 at 0:00
    
Maybe monad? en.wikipedia.org/wiki/Monad_(category_theory) –  alancalvitti Jan 8 '13 at 1:49

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up vote 9 down vote accepted

You can define category objects in a category with enough pullbacks. This is very useful and is called "internal category theory". The basics are explained in http://ncatlab.org/nlab/show/internal+category. More generally, one can define categories internal to a monoidal category by suitably replacing pullbacks by cotensors. This is explained in http://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category

Even more generally, but not written anywhere I know of, is the ability to define categories internal to operads with contensors. More general still, one can define operads internal to operads with cotensors.

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And even more general, we may define a category as a thing that satisfies the following formula: $\mathit{true}$ :-) –  Michal R. Przybylek Jan 8 '13 at 13:35
    
This reminds me a course in Software Engineering when the tutor drew a diagram: "object is in many-to-many relation with object" and claimed that it is the best possible entity-relationship diagram - it is purely abstract, hiding all irrelevant details, and what's more it is sound for every system; with only one drawback - it is completely useless... –  Michal R. Przybylek Jan 8 '13 at 14:18

I wouldn't say a category can be regarded as a set; in fact, it certainly can't unless it's small. But that's not what you're asking. I think the closest thing to what you're asking about is the notion of an internal category, which exists in all categories with pullbacks.

If $\mathcal{C}$ has pullbacks then an internal category in $\mathcal{C}$ is given by:

  • two objects $C_0, C_1$, which can be thought of as the 'object of objects' and 'object of arrows, respectively;
  • arrows $c, d : C_1 \rightrightarrows C_0$, which can be thought of as the 'codomain arrow' and 'domain arrow' respectively;
  • an arrow $i : C_0 \to C_1$, which can be thought of as the 'identity arrow arrow'; and
  • $m : C_1 \times_{C_0} C_1 \to C_1$, which can be thought of as the 'composition arrow'.

These arrows in $\mathcal{C}$ then interact according to how you'd expect them to according to the category axioms.

For more, see nCatLab.

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"I wouldn't say a category can be regarded as a set;" For example, the MacLane's book regards a category as a set which satisfies some algebraic conditions. –  Makoto Kato Jan 8 '13 at 0:32
    
@MakotoKato: This is an interesting foundational point. If you use ZF, or ZFC, as your foundation for category theory then, for example, there is no category of all sets and functions between them, since there would have to be a set of objects (i.e. a set of all sets) and hence the category would contain itself as an object. So when Mac Lane writes $\mathbf{Set}$ he isn't referring to the 'category' of all sets, which he calls a 'metacategory' instead. [Continued...] –  Clive Newstead Jan 8 '13 at 8:19
    
...but you can get over this foundational issue by interpreting the axioms of category theory in a different set theory (such as NBG) which has proper classes as objects, and not just as informal concepts. Then there really is a category of all sets and functions between them, but it is a class and not a set. Even if they say they don't, this latter approach is certainly how category theorists think about categories. –  Clive Newstead Jan 8 '13 at 8:21
    
...so it comes down to what you think a category is $-$ and when I talk about $\mathbf{Set}$ I want it to have all the sets as objects. –  Clive Newstead Jan 8 '13 at 8:24
    
Perhaps as you know, Mac Lane uses Grothendieck's universe. –  Makoto Kato Jan 8 '13 at 8:30

People have already mentioned that this is called an internal category and defined in the obvious way, so let me explain a couple of applications.

Theorem. Let $\mathcal{E}$ and $\mathcal{S}$ be elementary toposes and suppose $p : \mathcal{E} \to \mathcal{S}$ is a bounded geometric morphism. Then there exists an internal category $\mathbb{C}$ in $\mathcal{S}$ and a factorisation of $p$ as $$\mathcal{E} \longrightarrow [\mathbb{C}^\textrm{op}, \mathcal{S}] \longrightarrow \mathcal{S}$$ where $[\mathbb{C}^\textrm{op}, \mathcal{S}]$ denotes the topos of internal presheaves on $\mathbb{C}$ in $\mathcal{S}$, where the geometric morphism $\mathcal{E} \to [\mathbb{C}^\textrm{op}, \mathcal{S}]$ is a geometric inclusion.

Proof. See §B3.3 in Sketches of an elephant.

In other words, Giraud's theorem characterising Grothendieck toposes can be relativised to arbitrary base toposes – and in the strongest possible way, because we do not need to assume that $\mathcal{S}$ is cocomplete or boolean or anything like $\textbf{Set}$ at all.

Another application can be found in Janelidze's categorical Galois theory: under good conditions, the category of "split algebras" in one category will be (contravariantly) equivalent to the category of internal presheaves on an internal groupoid in another category. For example,

Theorem. Let $\sigma : R \to S$ be a Galois descent morphism of rings, and let $\textrm{Gal}(\sigma)$ be the corresponding Galois groupoid in the category of profinite spaces. Then the category of $R$-algebras split by $\sigma$ is contravariantly equivalent to the category of internal presheaves on $\textrm{Gal}(\sigma)$ in the category of profinite spaces.

Proof. See Theorem 4.7.15 in [Categorical Galois theory].

Also, I'll mention that Bénabou worked out how to define the notion of a locally internal category, which is rather more complicated. This can be found in §B2.2 of Sketches of an elephant.

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Sketches of an Elephant? Cool name! –  Haskell Curry Jan 8 '13 at 8:45
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@HaskellCurry. No, this is a nice name. It comes from a poem of "Molana", a famous Persian Poet. In that poem he say a story about some persons who encountered an elephant in darkness and every one were near part of that animal and by touching it every person thought different things and describe it in different details but when light comes they all see nobody was wrong but no one describe it complete. Author of this book want to say that all our objects are part of a more big family of objects. I suggest you to read it! both of them. –  AmirHosein SadeghiManesh Jan 8 '13 at 9:13

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