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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