[Math] Objects which can’t be defined without making choices but which end up independent of the choice

axiom-of-choicebig-listgm.general-mathematicslo.logic

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data, one first chooses some additional structure. And sometimes (often?) the constructed object a posteriori happens to be independent of the choice.

Examples which come to mind are the following:

  1. The trace of an endomorphism, defined by choosing a basis.

  2. Derived functors like $Ext$, $Tor$, defined by choosing projective/injective resolutions.

Now I have the feeling, that "natural" objects that are not dependent on choices should exist independently of these choices. So they should still exist in a universe where there's no way to choose and so I would expect that existence to be provable there. So I would expect there to be some way to avoid choosing anything in the first place in order to define such objects.
For instance, one can define the trace as the composite map:
$End_k(V)\cong V^*\otimes V \to k$ where $V^ *\otimes V\to k$ is evaluation and $V^ *\otimes V \to End_k(V)$ is given by $f\otimes v\mapsto (w\mapsto f(w)v)$.

So my questions are the following:

  1. Do you know examples of things which are natural in some sense but which can't be defined without choosing something first?
  2. Are $Ext$, $Tor$, etc. examples? i.e. is there a way to define derived functors without choosing resolutions?
  3. If things like in 1. exist, is there some way to make the statement precise that they can't be defined without choosing anything? Can such results be proven?
  4. Assuming again things like in 1. exist. Where exactly does the above informal "philosophical" argument fail? What is the deeper reason for the existence (or nonexistence) of such objects?

Best Answer

I think that the fundamental group of a path-connected space is an archetype of the phenomenon which you are describing. Its construction and definition requires a choice of basepoint ("Give me a place to stand, and I will move the world", as Archemedes said, quoted by Pappus), but it is independent of that choice up to automorphism (and it's only ever considered up to automorphism).

The "choiceless" object is the fundamental groupoid, but because groupoids are more complicated less familiar algebraic objects than groups, fundamental groups are much more commonly used... and they can't be defined without an arbitrary choice which they are independent of. We just need "a place to stand", that's all!

Philosophically, I think that the reason that we see examples of this phenomenon so often in topology (choice of basepoint, triangulation, smooth structure, lift, cell decomposition, Morse function, etc. where everything is independent of these choices) is that, by definition, manifolds (and, in a weaker sense, other commonly-studied classes of spaces such as CW complexes) "look the same" around any point. So in order to "nail down" a mathematical statement, sometimes we need to "stand somewhere", "introduce coordinates", "fix an arbitrary construction of the space"... and then we can translate our problem to a tractable and/or familiar algebraic category such as "groups", "modules", "symmetric monoidal something", or whatever. These "translations" are the subject of algebraic topology.

Related Question