Trivial objects show up in most every branch of mathematics, and we all know lots of examples: the trivial group, ring, vector space, module over a ring, graph, knot, homomorphism from one object to another, category, Boolean algebra, topology on a set, group representation, measure, and so on.
Trivial objects are so ubiquitous and fundamental that I strongly suspect that they form a sort of mathematical 'natural kind': that is, that there's some interesting and relatively simple property in virtue of which trivial objects count as trivial. But it's not immediately obvious (to me) what that property is.
A couple of ideas that seem on the right track, but don't quite work:
- "Trivial objects are small (i.e., supported on small sets)." Proexamples: The trivial group, ring, Boolean algebra, vector space, graph, etc. Counterexample: The trivial automorphism of your favorite big object.
- "Trivial objects are zero objects in their categories (or at least final objects)." Proexamples: the trivial group in Grp, the trivial ring in Rng, the trivial module over $R$ in $R$*-Mod*, the trivial vector space over $K$ in $K$*-Vect* (zero objects); the trivial ring in Ring, the trivial topological space in Top (final only). Counterxample: The trivial graph is not final (or initial) in Graph.
But I'm sure someone can do better! I'd much like to hear your ideas about what, if anything, trivial objects have essentially in common.
(I should say, by the way, that I'm not asking about things like trivial proofs or theorems, or trivial solutions to equations. The question is only about trivial objects, like the ones I named as examples in the first paragraph.)