No junk and no noise slogan

[Harry Chen]

Hi, Ken, excellent speech tonight for free. You mentioned that the free object has the properties of ‘no junk and no noise’. the initial algebra has similar slogan (http://homepages.feis.herts.ac.uk/~comqejb/algspec/node10.html) I wonder whether Initial object and free object have something in common?

Regards,

Harry

 

[Ben Hutchison]

On 24/03/2014 8:32 PM, "Harry Chen" <chen.ha...@gmail.com> wrote:
>
> Hi, Ken, excellent speech tonight for free.

+1!   Im regretting that we didnt record it. I reckon it would be the best explanation and demo of Free Monads available in the world ATM.

Ben

[Toby Corkindale]

Oh no! Makes it even worse that I missed it.
I was giving a talk elsewhere tonight, otherwise I would have liked to
catch your talk.

[Ishaaq Chandy]

+1 - Kudos on an excellent talk Ken!

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[Ken Scambler]

Thanks Ben! I wouldn't call it the best of anything in particular, but I think it added something to the current stable of tutorials - perhaps it would be useful for me to rework it into blog form.

Cheers, Ken

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[Ken Scambler]

Hey Harry,

I don't really know anything about "Initial Algebras", although according to Wikipedia, they do seem to be a kind of initial object. 

The language used in the article you link is almost identical to the language used to describe Free objects, so I wouldn't be surprised if they were the same thing, complete with a free/forgetful adjoint functors.   Great pickup!  There's so many great surprises in maths, where the patterns you see in one area turn up in a completely unrelated area later on.

Other than "no junk, no noise", the more rigorous definition of freeness involves a universal mapping property, which involves a "unique homomorphism", much like the Initial Algebra page you linked.

I'll add more detail about the UMP when I've got the time to check the Category Theory book again.

Cheers,
Ken

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[Harry Chen]

Hi, Ken, thanks for your reply.

  might I connect the two concepts(initial vs free) in this way?

  List (data type) is the initial object (I use ’the’ because the initial objects are up to isomorphic in a category) in the category of MONOID which is freely generated from any type. while free monad is the initial object in the category of MONAD which is freely generated from any Functors.

also I paste words excerpted from http://www.andrew.cmu.edu/user/avigad/Teaching/landc_notes.pdf (page 16):

"Steve Awodey tells me that category theorists think of freely generated inductively defined structures as having “no junk, no noise.” “No junk” means that there is nothing in the set that doesn’t have to be there, which stems from the fact that the set is inductively defined; and “no noise” means that anything in the set got there in just one way, arising from the fact that the elements are freely generated."

Regards

Harry

[Ken Scambler]

On 25/03/2014 1:21 PM, "Harry Chen" <chen.ha...@gmail.com> wrote:
>
> Hi, Ken, thanks for your reply.
>   might I connect the two concepts(initial vs free) in this way?
>   List (data type) is the initial object (I use ’the’ because the initial objects are up to isomorphic in a category) in the category of MONOID which is freely generated from any type.

Objects in MON are monoids; (List[A], ++, Nil) is one (freely generated from set A), but List[A] is the underlying set and not a monoid itself.  AFAICT (List[A], ++, Nil) is not initial either - is there a unique arrow (specifically, monoid homomorphism) to every other monoid? I would have thought the initial would be a trivial empty monoid or something like that.

while free monad is the initial object in the category of MONAD which is freely generated from any Functors.

It's probably analogous, whatever the answer is for monoids.

>
> also I paste words excerpted from http://www.andrew.cmu.edu/user/avigad/Teaching/landc_notes.pdf (page 16):
>
> "Steve Awodey tells me that category theorists think of freely generated inductively defined structures as having “no junk, no noise.” “No junk” means that there is nothing in the set that doesn’t have to be there, which stems from the fact that the set is inductively defined; and “no noise” means that anything in the set got there in just one way, arising from the fact that the elements are freely generated."

Other way around.

[Harry Chen]

Hi, Ken, thanks. I found this link(http://math.stackexchange.com/questions/58553/why-is-the-free-monoid-free) which might answer my question.

The free -object on X, when it exists, can be interpreted as an initial object, but in a different category

Thanks again for your time.

Harry