Would it be possible to propose a satisfying categorical definition for the notion of basis?

Question

I once came across a definition for the notion of basis that was independent of the type of the algebraic structure considered (although I cannot find where). Translated into category terminology, it went like this:

Let's consider a concrete category. We say that $B$ is a basis of object $X$ if $B=(B_i)_{i \in I}$ is a family of elements of $X$ such that for every object $Y$ and every family $(F_i)_{i \in I}$ of elements of $Y$, the function $f : B \rightarrow F$, defined by $$\forall i \in I,\quad f(B_i)= F_i$$ can always be extended to a unique homomorphism $\tilde f:X \rightarrow Y$.

To me it looks like such a definition describes most of the classical constructs we generally call bases. However a friend of mine considers this definition is not appropriate because it cannot take into account variants where we want to combine infinitely many elements (those cases are arguably not strictly algebraic). He offers two counter-examples: he considers that for the respective cases of Hilbert spaces and Banach spaces the appropriate equivalents of the notion of basis are respectively Hilbert bases and Schauder bases. Apparently in the latter case it is not even true that there necessarily exists a homomorphism mapping a given Schauder basis to another given Schauder basis. He suggests that the acceptable output families $F$ should be heavily restricted, but in the end he doubts it is even possible to come up with a working definition. His point is that those families should be too case-dependent to be easily expressed in terms of categorical terminology.

Has this question been tackled by category theory? Is my friend correct?

Answer 1

I'd say that in general, the right categorical notion of basis is given in terms of left adjoints of forgetful functors. Suppose you have a category $\mathcal C$ and a forgetful functor $F\colon\mathcal C \to \mathsf{Set}$. If it has a left adjoint $G\colon\mathsf{Set}\to \mathcal C$, then an object in $\mathcal C$ is free if it's in the image of $G$.

But in general it's not true that any two identifications $X\cong G(Y)$ and $X\cong G(Z)$ are equivalent. For example, if $R$ is a ring and $X$ is a finitely generated free $R$-module then it may have inequivalent bases under elementary substitutions. This phenomenon is measured by $K_1(R)$, which is $GL(R)/E(R)$, where $GL(R)$ is the infinite general linear group and $E(R)$ is the subgroup generated by the elementary matrices. You might also want to mod out by the units in $R$ if you want to allow scaling basis elements as well as elementary substitutions.

Schauder bases are not an example of this, especially since they come with an ordering, as convergence may be conditional. So you're not looking at a forgetful functor to sets at all in that case.

Answer 2

I'd say the definition you give is the best one. Things like Hilbert basis or Schauder basis aren't really basis, but more like special generating familly.

Note that your definition is exactly the local definition of "left adjoint to the forgetful functor set".

The problem with Hilbert basis, aren't the infinitary combinations — there are examples with infinitary combination that are handled very well by this notion:

Take $C$ to be the category of suplattices (preordered sets with arbitrary suprema) then the set of $\{x\}$ for $x \in X$ is a basis in your sense of the power set $\mathcal{P}(X)$. A general element of $\mathcal{P}(X)$ is a supremum of an arbitrary family of singletons.

Take $C$ to be the category of compact Hausdorff spaces, then for a set $X$, the principal ultrafilter $\delta_x$ for $x \in X$ form a "basis" of the Stone–Čech compactification (the space of all ultrafilters) of $\beta X$ of $X$. A general element of $\beta X$ is a limit (along an ultrafilter) of an infinite family of elements of $X$.

Ok, so what's wrong with say Hilbert Basis from this perspective? Well the point is that they aren't really basis, in the sense that they don't generate the space "freely".

The universal property for Hilbert basis involve as you said restriction on the type of family of element we consider. The simplest way to put it is to work in the category of Hilbert spaces and isometric maps and then if $(b_i)_{i \in I}$ is a Hilbert (orthonormale) basis of $\mathcal{B}$, then the data of a map $\mathcal{B} \to \mathcal{H}$ is the same as the choice of a family of elements of $t_i \in \mathcal{H}$ subject to the relation $\langle t_i,t_j\rangle = \delta_{i,j}$. (it is possible to express other universal properties using different class of maps than isometric maps, but this one is the simplest by far.)

This is completely analogous to the universal property you'll get for a presentation of a vector space by generator and relation: it means that $\mathcal{B}$ is freely generated by the familly $b_i$ subject to the relation $\langle b_i,b_j\rangle = \delta_{i,j}$.

A more general notion you can use that encompasses both is the notion of "generating family": a family of elements $b_i \in \lvert X\rvert$ (I'm using $\lvert X\rvert$ to denote the underlying set of $X$) is generating if for all object $Y$ the induced map

$$ \operatorname{Hom}(X,Y) \to \lvert Y\rvert^I $$

is injective.