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.
Best Answer
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.