From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8869 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: A construction for polynomials. Date: Mon, 04 Apr 2016 23:59:16 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1459937468 27174 80.91.229.3 (6 Apr 2016 10:11:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 6 Apr 2016 10:11:08 +0000 (UTC) To: Nikita Danilov , "categories" Original-X-From: majordomo@mlist.mta.ca Wed Apr 06 12:11:01 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ankQO-0005B3-H8 for gsmc-categories@m.gmane.org; Wed, 06 Apr 2016 12:11:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52001) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ankPr-0002Sf-7u; Wed, 06 Apr 2016 07:10:27 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ankPh-00045D-A0 for categories-list@mlist.mta.ca; Wed, 06 Apr 2016 07:10:17 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8869 Archived-At: Greetings, I think you've rediscovered, in > Nat(P, P) =3D Nat(Ring(Z[X], -), P) =3D 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) =3D Nat(Ring(Z[X], -)^n, P) = =3D Nat(Ring(Z[X_1, ..., X_n], -), P) =3D Z[X_1, ..., X_n]. Cheers, -- Fred ------ Original Message ------ Received: Mon, 04 Apr 2016 08:40:19 PM EDT From: Nikita Danilov To: 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) =3D Nat(Ring(Z[X], -), P) =3D 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" forgetfu= l > functor > from the co-slice category. For an object S of B define > = > R[S] =3D 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 =3D Ring, B =3D Set this gives the usual ri= ng 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 =3D= 1) > can > be composed, to each r:1->R and x:1->S corresponds a polynomial (provid= ed 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/ ]