Part 1/2

Sorry for elaborating the excursus, but wanted to give another stab to the profunctors vs linear algebra analogy trying to tie together what John has said. It's fun/profitable to pursue this, juicy, for me is work in progress. The gist of the idea is to compare the vector space monad coming from the free-forgetful adjunction relating sets and vector spaces, with something monad-like ensuing from the process of \\(\mathcal{V}\\)-enriched free **cocompletion**, relating \\(\mathcal{V}\\)-enriched categories and \\(\mathcal{V}\\)-enrichedly *cocomplete* categories.

The setup of the free-forgetful adjunction for sets and vector spaces is in example 2.4 [here](http://math.uchicago.edu/~may/REU2012/REUPapers/Sankar.pdf). Assuming a field \\(\mathbb{K}\\), the free vector space functor \\(F\\) takes a set \\(S\\) and produces the vector space with basis \\(S\\) made of formal linear combinations, or equally, finitely supported functions in \\(\mathbb{K}^S\\). The forgetful functor \\(U\\) preserves the set of formal linear combinations, but forgets vector space structure.

The adjunction produces a monad in \\(\mathbf{Set}\\), \\(T := U \circ F\\). E. Riehl describes the unit and multiplication in example 5.1.4(iii) of book ISBN 048680903X. Exercise 4.5.10 [here](https://math.feld.cvut.cz/ftp/velebil/downloads/cats-tacl-2017-notes.pdf) describes the Kleisli category as having sets as objects and \\(\mathbb{K}\\)-valued matrices as arrows.

In this monad there is a bit of "degeneracy" in that the Kleisli and EM-categories are equivalent and this needn't be the case in general.

From Riehl's book, section 5.2

> The upshot of Lemma 5.2.13 is that the Kleisli category for a monad embeds as the full subcategory of free T-algebras and all maps between such. Lemma 5.2.13 also tells us precisely when the Kleisli and Eilenbergâ€“Moore categories are equivalent: this is the case when all algebras are free. For instance, all vector spaces are free modules over any chosen basis, so the Kleisli and Eilenbergâ€“Moore categories for the free vector space monad are equivalent

Paul Taylor says:

> \\(\mathbf{Vsp}\\) (i. e. \\(\mathbf{Vect}\\)) is the Eilenberg-Moore category for the monad on Set induced by the adjunction in Example [7.1.4(b)](http://www.cs.man.ac.uk/~pt/Practical-Foundations/html/s71.html#r7.1.4) (our case). The Kleisli category consists of those vector spaces that have bases (which is all of them, given the axiom of choice).

If we drop Choice, for instance for constructivist reasons, that destroys the equivalence and the EM-category assumes the rol of the general category of vector spaces, and Kleisli's, the one for spaces with basis.

In particular, having bases, a map in the Kleisli category just says where the basis vectors of the source space are sent in the target one. Since linear maps preserve linear combinations, that defines the whole map between spaces. That's a case of [extension operator](https://en.wikipedia.org/wiki/Kleisli_category#Extension_operators_and_Kleisli_triples) in general monad theory.

Sorry for elaborating the excursus, but wanted to give another stab to the profunctors vs linear algebra analogy trying to tie together what John has said. It's fun/profitable to pursue this, juicy, for me is work in progress. The gist of the idea is to compare the vector space monad coming from the free-forgetful adjunction relating sets and vector spaces, with something monad-like ensuing from the process of \\(\mathcal{V}\\)-enriched free **cocompletion**, relating \\(\mathcal{V}\\)-enriched categories and \\(\mathcal{V}\\)-enrichedly *cocomplete* categories.

The setup of the free-forgetful adjunction for sets and vector spaces is in example 2.4 [here](http://math.uchicago.edu/~may/REU2012/REUPapers/Sankar.pdf). Assuming a field \\(\mathbb{K}\\), the free vector space functor \\(F\\) takes a set \\(S\\) and produces the vector space with basis \\(S\\) made of formal linear combinations, or equally, finitely supported functions in \\(\mathbb{K}^S\\). The forgetful functor \\(U\\) preserves the set of formal linear combinations, but forgets vector space structure.

The adjunction produces a monad in \\(\mathbf{Set}\\), \\(T := U \circ F\\). E. Riehl describes the unit and multiplication in example 5.1.4(iii) of book ISBN 048680903X. Exercise 4.5.10 [here](https://math.feld.cvut.cz/ftp/velebil/downloads/cats-tacl-2017-notes.pdf) describes the Kleisli category as having sets as objects and \\(\mathbb{K}\\)-valued matrices as arrows.

In this monad there is a bit of "degeneracy" in that the Kleisli and EM-categories are equivalent and this needn't be the case in general.

From Riehl's book, section 5.2

> The upshot of Lemma 5.2.13 is that the Kleisli category for a monad embeds as the full subcategory of free T-algebras and all maps between such. Lemma 5.2.13 also tells us precisely when the Kleisli and Eilenbergâ€“Moore categories are equivalent: this is the case when all algebras are free. For instance, all vector spaces are free modules over any chosen basis, so the Kleisli and Eilenbergâ€“Moore categories for the free vector space monad are equivalent

Paul Taylor says:

> \\(\mathbf{Vsp}\\) (i. e. \\(\mathbf{Vect}\\)) is the Eilenberg-Moore category for the monad on Set induced by the adjunction in Example [7.1.4(b)](http://www.cs.man.ac.uk/~pt/Practical-Foundations/html/s71.html#r7.1.4) (our case). The Kleisli category consists of those vector spaces that have bases (which is all of them, given the axiom of choice).

If we drop Choice, for instance for constructivist reasons, that destroys the equivalence and the EM-category assumes the rol of the general category of vector spaces, and Kleisli's, the one for spaces with basis.

In particular, having bases, a map in the Kleisli category just says where the basis vectors of the source space are sent in the target one. Since linear maps preserve linear combinations, that defines the whole map between spaces. That's a case of [extension operator](https://en.wikipedia.org/wiki/Kleisli_category#Extension_operators_and_Kleisli_triples) in general monad theory.