TITLE: Objects which can't be defined without making choices but which end up independent of the choice
QUESTION [35 upvotes]: 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:  

The trace of an endomorphism, defined by choosing a basis.
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:
 Do you know examples of things which are natural in some sense but which can't be defined without choosing something first?
 Are $Ext$, $Tor$, etc. examples? i.e. is there a way to define derived functors without choosing resolutions?
 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?
 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?

REPLY [3 votes]: In topology, many maps are only characerized by homotopy properties. Proving that a certain homotopy class exists often requires the construction of an actual element in it, and this element is typically not uniquely determined. Aa advanced jargon phrase is that a ''construction depends on a contractible space of choices''. Here are two examples.
If $P \to X$ is a $G$-principal bundle, a ''classifying map'' $f: X \to BG$ is characterized by the property that $f^{\ast} EG \cong BG$, and this is unique up to homotopy. What one does is to prove that the space of $G$-equivariant maps $g:P \to EG$ is contractible (if $X$ is paracompact). Any such $g$ descends to a classifying map $X \to BG$ (and any classfying map is covered by such a $g$). Therefore, $f$ depends on a contractible space of choices and is therefore unique up to homotopy in the strongest possible sense. The concrete construction depends on data such as local trivializations of the bundle and partitions of unity and is therefore not unique or ''natural'' or ''canonical''.
The other example I want to mention is the Pontrjagin-Thom isomorphism, both of whose direction depend on choices. It states a bijection of the set of bordism classes of framed $n$-manifolds in $R^{m+n}$ and homotopy classes of maps $S^{m+n} \to S^m$.
Only after descending to discrete invariants (homotopy and bordism groups), the construction is well-defined. Passing from a bordism class to a homotopy class requires choosing an embedding of the manifold into a euclidean space and the choice of a tubular neighborhood and the choice of a concrete framing. In the reverse direction, you need a representative of the homotopy class by a smooth map and a regular value.