nLab too simple to be simple

Idea

There is a general principle in mathematics that

A trivial object is too simple to be simple.

Quite often, classical references will define ‘simple’ (or an analogous term) in naïve way, so that a ‘trivial’ object is simple, but later it will become clear that more sophisticated theorems (especially classification theorems) work better if the definition is changed so that the trivial object is not simple. Usually this can be done by changing ‘if’ to ‘iff’ (or sometimes changing ‘or’ to ‘xor’) in the classical definition.

The general pattern is a progression of definitions (of ‘simple’) from more to less naïve:

1. Imprecise summary: there are no foos. (Example: in a field, every element is invertible.)
2. Classical definition: there are no nontrivial foos. (Example: in a field, every element other than $0$ is invertible.)
3. Modern definition: there are no nontrivial foos, but there is the trivial foo. (Example: in a field, every element other than $0$ is invertible, and $0$ is non-invertible.)

This should be distinguished from barring the trivial object entirely. Historically, many people wanted to say that the empty set is not a set, and some people still say that the trivial ring is not a ring or that the improper filter is not a filter. This allows one to state later definitions (as of field or ultrafilter) without having to exclude the trivial object, but this does the exclusion too early.

Examples

Examples include:

• 1 is not a prime number.
• The trivial ring is not a field (or even an integral domain); though see possibly trivial field and possibly trivial integral domain
• The trivial group is not a simple group (which is the trope-namer).
• A zero object is not a simple object (generalizing the previous example).
• The empty space is not a path-connected space (or even a connected space).
• The improper ideal is not a maximal ideal (or even a prime ideal).
• The improper filter is not an ultrafilter.
• An empty function is not a constant function.
• A bottom element is not an atom.

But of course one may still find definitions used that disagree (see discussion at connected space and empty space, for example).

Perhaps the basic example is that

• False is not true.

Nobody would be naïve enough to believe otherwise in this case, of course. The point with this example is that, in the other cases, the naïve definition imposes only a uniqueness requirement (so that some set of possibilities —such as the set of proper divisors of a prime number, or the set of non-invertible elements of a field— must be a subsingleton) when it should in fact impose an existence and uniqueness requirement (so that the set of possibilities must be a singleton). With truth values, uniqueness is automatic, so existence is easier to notice. More abstractly, the naïve definition is about $(-1)$-truncation, while the more sophisticated definition is about $(-2)$-truncation, which is more often relevant.

Relationship to biased definitions

In many of the above examples one can obtain the sophisticated definition from the naïve definition by replacing a $2$-ary function by a function of arbitrary (finite) arity. For example we would replace

• $n$ is prime if whenever $n = a b$ we have $n = a$ or $n = b$

with

• $n$ is prime if whenever $n = \prod_{i=1}^k a_i$ we have $n = a_i$ for some $i$

Then $1$ is not a prime because it is equal to the empty product ($k = 0$) but not equal to any of the $a_i$ (because there aren’t any)! Similarly we have:

• The empty space is the empty union, and therefore not connected.
• In the trivial ring the empty product is equal to zero, and so this ring is not integral.
• False is the empty disjunction, and hence is not true.

This illustrates one advantage of using unbiased rather than biased definitions: if one has replaced “for all $n$” with “for both $0$ and $2$” then it is very easy to forget the $0$ case and end up with a definition that fails for the trivial case. (Unfortunately, it seems to be easy to forget $0$ even when using unbiased definitions, but hopefully less easy.)

In a similar vein we can define path connected by

• A space is path connected if for each (Kuratowski)-finite subset there is a path (a continuous map from the unit interval) passing through every point of that subset.

Then the empty space is not path connected because it has no paths at all and hence no path through the empty subset.