Re: A construction for polynomials.

[Anders Kock <kock@math.au.dk>]

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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