[Haskell-cafe] Endo a, endomorphisms

[Tom Ellis]

On Sun, Dec 05, 2021 at 02:39:01PM +0000, Adrian wrote:
> On Sunday, December 5th, 2021 at 8:13 AM, Tom Ellis <tom-lists-haskell-cafe-2017 at jaguarpaw.co.uk> wrote:
> > On Sun, Dec 05, 2021 at 02:02:09PM +0000, Adrian via Haskell-Cafe wrote:
> > > In the Data.Monoid library there is a monoid type instance for
> > > Endo a that is described in the following manner: > The monoid
> > > of endomorphisms under composition.  I understand how it is a
> > > monoid of endofunctions, but am unclear on how it is a monoid of
> > > endomorphisms.
> >
> > I'm not sure what you mean. The morphisms in this case are exactly
> > functions. "Endomorphism" and "endofunction" have the same meaning in
> > this context.
> 
> According to Algebra [Hungerford 74], an endomorphism is an
> endofunction that is a homomorphism.  A set of endomorphisms is
> quite distinct from a set of endofunctions in this regard.

The "endomorphism" terminology in use in this case comes from category
theory rather than algebra narrowly:

> An endomorphism of an object x in a category C is a morphism f: x → x

https://ncatlab.org/nlab/show/endomorphism

In the case of the Endo documentation, the category under
consideration is Hask, the category whose objects are Haskell types
and whose morphisms are Haskell functions.  Therefore "endofunction"
and "endomorphism" refer to the same thing in this context.