From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8867 Path: news.gmane.org!not-for-mail From: Nikita Danilov Newsgroups: gmane.science.mathematics.categories Subject: A construction for polynomials. Date: Mon, 4 Apr 2016 16:32:43 +0300 Message-ID: Reply-To: Nikita Danilov NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1459816620 21150 80.91.229.3 (5 Apr 2016 00:37:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 5 Apr 2016 00:37:00 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Apr 05 02:36:52 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 1anEzE-0003BZ-FC for gsmc-categories@m.gmane.org; Tue, 05 Apr 2016 02:36:52 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51434) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1anEyL-0006HX-EO; Mon, 04 Apr 2016 21:35:57 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1anEyD-0006Bk-KB for categories-list@mlist.mta.ca; Mon, 04 Apr 2016 21:35:49 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8867 Archived-At: 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/ ]