# Category whose objects and morphisms are the same structures

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

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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