12
$\begingroup$

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?

$\endgroup$
4
  • 8
    $\begingroup$ The main question, which I read as essentially “What category-theoretic notions of basis have been studied?”, is a good one, and I hope will get good answers by people who know the topic well. But your friend’s opinion that the notion you give is “not appropriate” is rather subjective, and not really either “correct” or incorrect — as you note, this notion subsumes many classical notions of basis, but fails to capture other notions. So it’s appropriate for some purposes, but not all. $\endgroup$ Apr 23, 2023 at 10:48
  • 2
    $\begingroup$ I am not an expert, but it looks to me like this notion has a good chance of being appropriate whenever the forgetful functor from your category to the category of sets-and-functions "has good behaviour" in some sense. For instance k-Vect is monadic over Set, if I recall correctly. I am not sure if the notion above is good for R-Mod when R is a more general ring, since "minimal generating sets need not be independent" (szyzygies etc) $\endgroup$
    – Yemon Choi
    Apr 23, 2023 at 12:47
  • 1
    $\begingroup$ It bears stating explicitly that the definition you gave is simply the definition of a free object over a set of generators. (The answers sort of hint at this, but it should be stated explicitly, since it seems like the OP is not aware of this.) $\endgroup$ Apr 24, 2023 at 13:42
  • $\begingroup$ You might be interested in this paper, which is a different category-theoretic definition of basis. (I've only skimmed it and noted that it sounded interesting - I don't know how it relates to your idea.) Bart Jacobs - Bases as Coalgebras (2013). $\endgroup$
    – N. Virgo
    Apr 25, 2023 at 9:09

2 Answers 2

17
$\begingroup$

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.

$\endgroup$
5
  • 1
    $\begingroup$ Is "forgetful functor" an informal term, or is there some formal way to define when a functor is forgetful? Or is it a datum that is to be specified together with the category? $\endgroup$
    – LSpice
    Apr 23, 2023 at 16:49
  • 2
    $\begingroup$ I recommend the Wikipedia page on forgetful functors. $\endgroup$ Apr 23, 2023 at 17:08
  • 1
    $\begingroup$ Re, as far as I can tell, although wiki:forgetful functor gives examples of forgetful functors, and properties they "tend to" have, it doesn't actually define what a forgetful functor is, only what it does. More precisely, it suggests a definition via universal algebra ("curtailing the signature"), but I don't see how to use it to recognise when a functor between abstract categories (that I do not know a priori consists of, say, algebras of a given signature) is, or is not, forgetful. $\endgroup$
    – LSpice
    Apr 23, 2023 at 18:32
  • $\begingroup$ I guess I agree. Even MacLane uses the term in "Categories for the Working Mathematician" via examples. $\endgroup$ Apr 23, 2023 at 19:02
  • 4
    $\begingroup$ I think whether a given functor is "forgetful" really does depend on signatures or presentations of the categories involved. For any functor at all, we can replace its domain by an equivalent categories so as to make it look forgetful. For instance, the category of sets is equivalent to the category of vector spaces equipped with a basis and linear maps that preserves these bases; the free functor from this category of "sets" to the category of vector spaces is then simply "forgetting the bases". $\endgroup$ Apr 25, 2023 at 20:23
9
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ I'm very curious about this unexpectedly simple definition for generating families. Could we leverage it to define "minimally generating families" ? (= for which any subfamily is not generating anymore). Would that be equivalent to the definition for a base proposed in the OP or would we get a more general concept ? $\endgroup$ May 5, 2023 at 10:07
  • 1
    $\begingroup$ Interesting, I think the notion of minimal generating familly generalize the notion in the question, and includes things like Hilbert basis... $\endgroup$ May 6, 2023 at 8:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.