by Musa Al-hassy
Last Updated September 21, 2018 20:20 PM

Warning: I know little linear algebra and my assertions below may all be incorrect. I am interested in lists --i.e., free monoids-- and my interest has led me to [finite-dimensional] vector spaces.

Let π be any field, then we can freely equip any [finite] set S with a π-vector space $F(S) = (S β π)$, a set of functions where vector addition is performed pointwise, and likewise for the other vector operations. Moreover any function $S β |π±|$ from $S$ to the underlying set of a vector space π± can be extended to a linear operator $F(S) β π±$ that behaves the same on elements of $S$ construed as vectors of $F(S)$.

That is, $F$ is a βfree functorβ: It constructs the least vector space that contains a (copy of a) given set.

It is known that every vector space π₯ admits a basis Ξ² and so is isomorphic to $F(Ξ²)$ --which is essentially the dual space of π₯. That is, every vector space is free; i.e., is in the image of functor $F$.

Of-course we have to *choose* a basis to realise a vector space as a free object; there is
no canonical basis.

( Aside: With this in-hand, we can easily prove equi-dimensional spaces are necessarily isomorphic: That they're equi-dimensional means their basis are in bijection, but free objects are unique up to unique isomorphism and so their F-images must be unique up to unique isomorphism as well. That is, $π₯ β π^{dim π±} = π^{dim π±} β π¦$. Neat stuff! )

Questions:

*Intuitively*why is it that vector spaces admit basis.- I'm not interested in a proof.
- For example, why is it that monoids or rings are not always generated by some set, but vector spaces are. The definition of a vector space does not immediately give rise to a basis.

All [finite dimensional] vector spaces admit basis and so the objects of π½π¬πͺ, the category of finite dimensional vector spaces and linear operators, are all images of a free functor. (Assuming we have a way to

*choose*basis.)- Is there a name for categories whose objects are all isomorphic to an image of a free functor?
- What other examples of such categories are there?
- Are there any constructions that produce such categories.

Lists over a set $S$ are isomorphic to $β_{n β β} SβΏ$ and provide the

*free monoid*over a given set $S$. Why is it that lists, whence monoids, are far more prevalent than their vector space counterparts within computing science.

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