Is there a useful limit or co-limit of a diagram that has only a single object?

Question

I'm starting to study category theory kind of informally and everytime I read about the definitions of limits and co-limits, the first three examples are always the same:

• terminal/initial objects,
• products/coproducts,
• and pullbacks/pushouts.

It's always explained how terminal (initial) objects are the limit (colimit) of a an empty diagram and products (coproducts) are the limit (colimit) of a diagram with only two objects with no additional structure.

I always get that feeling of "what about the diagram with a single object?", but since that's never mentioned, I figured it must be something trivial or not useful. In a Poset it feels it would be some kind of lowest upper bound/greatest lower bound of a single object or something like that, but that looks like it would be trivially equal to the object itself. So, first question: is this limit interesting or is it trivial or maybe not used anywhere?

A related question: Recently I started reading studying the book Mathematical Physics by Robert Geroch (*) and when he defines a free group it feels a lot like what the limit of a diagram with a single object would be: "a free group on the set $S$ is a group $G$ together with a mapping $\alpha$ from $S$ to $G$ such that for any other group $G'$ with a mapping $\alpha'$ from $S$ to $G'$, there exists a unique homomorphism $\mu: G \to G'$ such that the diagram commutes", meaning that $\alpha' = \mu \cdot \alpha$. But of course that's not a limit because G and S belong to different categories and $\alpha$ and $\alpha'$ are not morphisms. The books defines other kinds of free things after that in a similar way -- free vector spaces, free topological spaces, etc. So, the second question is: are those free constructions related to limits? Is there some category where those constructions are actual limits of a diagram with a single object?

(*) I'm a physicist and my mathematical training was always focused on mathematical methods for physics and very little in formal, rigorous mathematics - since this book doesn't assume a lot of education in things like topology, algebraic geometric, etc, it feels more adequate for my background. Would love other suggestions.

Answer 1

One example: Let $X$ be a set with a group $G$ acting on it. Consider the diagram, in the category of sets, with $X$ as its only object, but with all the elements of $G$ (considered as permutations of $X$) as morphisms. The limit of this diagram is the subset of $X$ consisting of the points fixed by the group action, and the colimit is the set of orbits of the action.

Answer 2

Let $|G|$ be set of elements of a group $G$. Similarly, if $f : G \to H$ is a group homomorphism, let $|f|$ be the underlying mapping on elements. A diagrammatic translation of that definition of free group is as follows.

Consider the category whose objects are:

• Pairs $(G',\alpha')$ consisting of a group $G'$ together with a function of sets $\alpha' : S \to |G'|$
• Arrows $(G',\alpha') \to (G'', \alpha'')$ are group homomorphisms $\beta : G' \to G''$ satisfying $|\beta| \circ \alpha' = \alpha''$

It turns out this category has an initial object (which can be seen either as the colimit of an empty diagram or the limit of the entire category), and the pair $(G,\alpha)$ is taken to be an initial object.

So, I think your intuition was mashing together two separate facts:

• The free group is indeed the colimit of a fairly small diagram — the empty one!
• All of the objects in this category involve maps from the single set $S$

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Incidentally, this sort of pattern comes up a lot — constructing a category whose objects are collections of interesting data, and then doing things in that category.

The particular category above is an example of a comma category from the constant functor $* \to \mathbf{Set}$ whose image is $S$ to the forgetful functor $\mathbf{Grp} \to \mathbf{Set}$. In fact, one of the things you can do with comma categories is precisely to make sense of arrows between objects of different categories.

The construction above is also an example of an adjunction; the "free group" functor $\mathbf{Set} \to \mathbf{Grp}$ is the left adjoint to the "underlying set of elements" functor $\mathbf{Grp} \to \mathbf{Set}$.