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Are there important or interesting (nontrivial) examples of categories where the objects and morphisms are the same structures (e.g. all sets, or all functions)?

For instance, consider the construct $\mathbf{C}$ whose objects are functions and morphisms are functions, defined by $\hom_\mathbf{C}(f, g) = \hom_\mathbf{Set}(\operatorname{dom}(f), \operatorname{dom}(g))$. The forgetful functor $U : \mathbf{C} \to \mathbf{Set}$ that maps objects to their domain and morphisms to themselves.

This is a rather trivial and uninteresting example. Are there more interesting examples in research?

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I do not understand what you are asking. You want examples of what, exactly? – Mariano Suárez-Alvarez Nov 5 '12 at 1:29
Small category? – M.B. Nov 5 '12 at 1:30
Well, every function is a set, so... – Kevin Carlson Nov 5 '12 at 1:32
I've added an example to illustrate what I meant by "same structure", I hope that clears it up. – Herng Yi Nov 5 '12 at 1:41
I don't understand your example -- given two objects (functions) $f$ and $g$, what is $\hom_\mathbf{C}(f,g)$? – Brad Nov 5 '12 at 2:18
up vote 3 down vote accepted

Here's a slightly silly example: for any group $G$, there is a category $\mathbb{E} G$ whose objects are the elements of $G$, and we define $\mathbb{E} G (g_1, g_2)$ to be the singleton $\{ h \in G : h g_1 = g_2 \}$. Composition is induced by the group operation of $G$. Thus both the objects and morphisms of $\mathbb{E} G$ are elements of $G$.

Where does this example come from? Well, there is another category associated to $G$, called $\mathbb{B} G$, and this is just $G$ thought of as a one-object category. We can form the category $[\mathbb{B} G, \textbf{Set}]$ of all functors $\mathbb{B} G \to \textbf{Set}$ and if you think about it for a little while you will see that this is just the category of sets equipped with a left $G$-action. Since $G$ acts on itself, it can be regarded as an object of this category. On the other hand, there is a construction that takes functors $P : \mathbb{B} G \to \textbf{Set}$ to special functors $\pi_P : \mathbb{E} P \to \mathbb{B} G$, known variously as the "category of elements" or the "Grothendieck construction". The category $\mathbb{E} G$ is the result of this construction applied to the $G$-set $G$.

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+1 this is an excellent example! – magma Nov 5 '12 at 10:02

You can always identify objects with their identity morphisms. So, in every category, there are only morphisms.

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You can have a functor category, where the objects are functors and the morphisms are natural transformations.

I'm not sure exactly what you mean by "same objects" though.

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