A construction for polynomials.

[Nikita Danilov <danilov@gmail.com>]

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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