A construction for polynomials.

A construction for polynomials.

[Nikita Danilov]

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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Re: A construction for polynomials.

[Anders Kock]

Dear Nikita,

the special significant case with which you begin, deals with the forgetful  
functor (which you call P) from rings to sets. It lives in a category 
(topos) E of covariant functors from rings to sets, and this category you 
do not give a name. But P and E seem to me to be the main actors in your 
construction. What you call R[S] (for R in Rings, and S in Sets) is the 
value at R of the exponent object P^S -> P in E. In particular, R[1] is 
the value at R of the object P -> P in E. What you observe has as a 
special case the fact that P -> P, as an object in E, has the universal 
property of "P[X]", the free P-algebra in one generator; i.e.
P[X] = (P -> P) 
in E. This coincidence, of a colimit type universal property (of P[X]), with 
a limit type universal property (of the exponential P -> P), is 
significant, and generalizes to further duality results in E. 
It also generalizes to any othe algebraic theory T, not just to the theory 
of rings. For instance, for T the initial algebraic theory (the algebraic 
theory of sets), it implies that P+1 = (P -> P).
For an elaboration of these generalizations, see my "Duality for Generic 
Algebras", Cahiers 56 (2015), 2-14, or 
http://home.math.au.dk/kock/DGA03.pdf

Anders


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Re: A construction for polynomials.

[Rory Lucyshyn-Wright]

Dear Nikita,

The constructions that you describe can be understood as variations on
Lawvere's concept of the _algebraic_structure_ of a (Set-valued)
functor:

   F. W. Lawvere, Functorial semantics of algebraic theories,
    Dissertation, Columbia University, New York, 1963.
    Available in: Repr. Theory Appl. Categ. 5 (2004).
    Chapter III, Section 1.

The first construction that you describe is an instance of Lawvere's
notion, wherein the algebraic structure of the functor P is the
algebraic theory T of commutative rings.  The hom-set Nat(P,P) ~= Z[X]
that you describe is one of the hom-sets of this algebraic theory T,
namely the set of unary operations, which underlies the free T-algebra
on one generator.

In the case where we take B = Set, your second construction is an
instance of Linton's formulation of the _equational_structure_ of a
Set-valued functor as defined in

   F. E. J. Linton, Some aspects of equational categories,
     Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965),
     Springer, 1966, pp. 84-94.
     Section 2.

This notion of equational structure is an adaptation of Lawvere's
notion to the setting of Linton's _equational_theories_, which are
algebraic theories in which the arities are allowed to be arbitrary
(small) sets rather than just finite cardinals.  Indeed, take U to be
the relevant functor R/C --> B = Set and consider the equational
structure T of U in Linton's sense.  Given a set S, the set of natural
transformations that you describe is one of the hom-sets of the
resulting equational theory T, namely the (possibly large) set of all
S-ary operations, which (if it is small) underlies the free T-algebra
on the set S.

In the general case, where B is an arbitrary cartesian closed
category, your second construction is closely related to (but not an
instance of) Dubuc's formulation of the algebraic structure of a
(tractable) V-valued V-functor, as defined in

   E. J. Dubuc, Enriched semantics-structure (meta) adjointness,
    Rev. Un. Mat. Argentina 25 (1970), 5-26.
    Section 4.

This notion of enriched algebraic structure is a generalization of
Linton's above notion of equational structure (and so in turn is an
adaptation of Lawvere's notion) to the context of the V-theories of
Dubuc, which are V-enriched algebraic theories in which the arities
are arbitrary objects of V (so that in the case V = Set, Linton's
notion is recovered).  As Dubuc demonstrates, this notion of algebraic
structure is very closely related to the notion of codensity monad.

The divergence between your second construction and Dubuc's
formulation lies in the fact that you use an ordinary, non-enriched
V=B-valued functor rather than a V-functor and, correspondingly, you
form a set of natural transformations rather than an object of
V-natural transformations.

Note that above we have made no special use of the coslice category
R/C, so this aspect can be analyzed separately.

Best wishes,
Rory

-----Original Message-----
From: Nikita Danilov
Sent: Monday, April 04, 2016 10:32 AM
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.


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]



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Re: A construction for polynomials.

[Fred E.J. Linton]

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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