Re: A construction for polynomials.

["Fred E.J. Linton" <fejlinton@usa.net>]

Greetings,

I think you've rediscovered, in

>      Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X],

the unary fragment of what Lawvere called the algebraic structure 
of that functor Rings --> Sets.

The n-ary fragment resides in the analogous calculation

Nat (P^n, P) = Nat(Ring(Z[X], -)^n, P) 
  = Nat(Ring(Z[X_1, ..., X_n], -), P) = Z[X_1, ..., X_n].

Cheers, -- Fred

------ Original Message ------
Received: Mon, 04 Apr 2016 08:40:19 PM EDT
From: Nikita Danilov <danilov@gmail.com>
To: <categories@mta.ca>
Subject: categories: A construction for polynomials.

> Dear list,
> 
> I am looking for a reference to the following construction.
> 
> In the simplest case, consider the forgetful functor P:Ring->Set, from the
> category of commutative rings with a unit. Then the set of natural
endomorphisms
> Nat(P, P) can be identified with the set of polynomials of one variable
with
> integer coefficients Z[X]. This can be easily seen by the chain of
> isomorphisms
> 
>      Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X],
> 
> where the first isomorphism is due to Z[X] being a free ring with one
> generator
> and the second is by Yoneda's lemma. Alternatively, just observe that
> polynomial
> functions are precisely ones commuting with every ring homomorphism.
> 
> In general, let P:C->B be a functor from an arbitrary C to a cartesian
> closed
> B. Select an object R in C and let R/P:R/C->B be the "obvious" forgetful
> functor
> from the co-slice category. For an object S of B define
> 
>      R[S] = Nat(R/P * hom(S, -), R/P),
> 
> where hom is the internal hom functor of B and * is functor composition  in
> the
> diagrammatical order. For C = Ring, B = Set this gives the usual ring of
> polynomials with coefficients in R and variables from S.
> 
> This construction extends to a functor C x B -> Set and has some nice
> properties
> of the usual polynomials: polynomials of "one variable" (i.e., when S =  1)
> can
> be composed, to each r:1->R and x:1->S corresponds a polynomial (provided P
> maps
> 1 to 1).
> 
> Has this or dual (where C/R is used instead of R/C) construction been
> studied?
> Maybe in enriched contexts?
> 
> Thank you,
> Nikita.
> 


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