# What are trivial objects, in general?

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:

1. "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.
2. "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.)

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Objects whose existence is left as an exercise to the reader. :-) –  Asaf Karagila Feb 21 at 6:06
Can't "trivial proofs and theorems" be considered trivial objects in a suitable category? I don't know the technical details very well, but my intuitive understanding is that some construction related to syntactic categories leads to a notion of theorems or proofs as objects/morphisms in a category, in which there should be trivial objects (such as theorems with empty conclusion or conclusion identical to a hypothesis). –  Daniel Hast Feb 21 at 6:19
Sometimes it is also pretty ambiguous. I think a lot of people call a representation $G \to \mathrm{Aut}(V)$ trivial as soon as $G$ acts trivial (the image of the map is $\{\mathrm{id}_{V}\}$). However, in light of what you say, one should only call the $0$-dimensional representation trivial. (I admit that I sometimes get confused which of the two 'trivials' is meant, when someone concludes that a particular representation is trivial.) –  jmc Feb 21 at 8:04
initial + final? –  Fernando Muro Feb 23 at 21:35

When discussing family resemblance, Wittgenstein emphasized the example question

What is a game?

which he argued has no definite answer. Mathematically, the point is simply that we can have $$A\cap B\ne\varnothing,\quad B\cap C\ne\varnothing, \quad A\cap C\ne\varnothing,\quad \text{and yet}\quad A\cap B\cap C=\varnothing.$$

In your case, some examples of adjectives $A, B, C,\dots$ pertaining to a trivial object may be

• Extreme (they are the smallest or largest in some order; not just minimal or maximal)
• Unique (so an ultrafilter is probably not trivial)
• Definable (so a principal ultrafilter might qualify)
• Of low Kolmogorov complexity (having a short definition)
• Always the same in different contexts (for example, the trivial automorphism of any structure)
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The 'mere family resemblance' possibility wasn't lost on me (I'm a philosopher!), and you may well be right that that's the best we can do in this case. At any rate, these are interesting and helpful partial characterizations. I wonder especially how far one could run with Kolmogorov complexity -- are there any obvious counterexamples to the claim that trivial objects have low complexity? –  WBlase Feb 21 at 17:15

I think nearly everything I call trivial is the initial or terminal object in some category (but not necessarily a zero object, as many familiar categories don't have zero objects). Let's go through all of your examples to verify this:

the trivial group, ring, vector space, module over a ring, graph, category, Boolean algebra, group representation

Terminal objects. (You claim that the trivial graph is a counterexample, but with the correct definition of the category of graphs this is not true.)

homomorphism from one object to another

Depending on the objects, there many not be any such morphisms. I guess you refer to zero morphisms. When they exist in a category $C$, I think they should be something like the initial object in the category of functors $C^{op} \times C \to \text{Set}$ and extranatural transformations.

topology on a set

Which topology is the trivial topology? Either the discrete or the indiscrete topology could qualify here (the terminal and initial objects respectively in the poset of topologies on a set, ordered by inclusion).

measure

Initial object in the poset of measures, ordered by pointwise inequality.

The trivial automorphism of your favorite big object.

Initial object in the category of pairs (a group $G$, a homomorphism $G \to \text{Aut}(X)$).

The only example you give that doesn't easily fit into the above classification is

knot

but in this case knots don't obviously form a category so one has to appeal to other notions here. For example, the unknot (I don't like the term "trivial knot" because it sounds like it could refer to the empty link) is the identity element with respect to knot sum, and initial and terminal objects are identity elements with respect to coproduct and product, respectively, so there's certainly a familial resemblance here. Trivial automorphisms can also fit here.

One categorical structure that knots fit into is called the tangle category. In this category knots are morphisms instead of objects and I don't see a clear categorical description of the unknot in this context. I would avoid the term "trivial knot" for this reason but would be happy with the term "trivial link" to mean the empty link.

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