Category theoretic description of evaluation of polynomials

Question

How does one describe the evaluation of polynomials category theoretically?

We have some $f\in K[x_1,\cdots,x_n]$ and some $(k_1,\cdots,k_n)\in k^n$ and we take $f(k_1,\cdots,k_n)=r$ where $r\in \Bbb F$ for some field $\Bbb F$.

So we have $f\in \bf{Cat}(Kalg)$ and then some object $r$ in some other category?

What categories are we working in, and what is the functor between them, else, are we working in the one category and these are given by some morphism?

Answer 1

While I don't know if this is a complete answer to your question, there is a concept of evaluation map in the definition of exponential objects in a category. In a category with sets as objects and polynomial functions as morphisms the corresponding eval map may be what you want.

Answer 2

Fix a base field $k$. Morphisms

$$\varphi : k[x_1, \dots x_n] \to A$$

from $k[x_1, \dots x_n]$ to any other $k$-algebra $A$ correspond to $n$-tuples of elements of $A$, as follows: if $(a_1, \dots a_n)$ is an $n$-tuple, the corresponding morphism is the evaluation morphism

$$\varphi_{(a_1, \dots a_n)} : k[x_1, \dots x_n] \ni f(x_1, \dots x_n) \mapsto f(a_1, \dots a_n) \in A.$$

So from this point of view, evaluation is the way to make explicit the universal property of the polynomial algebra. There are various other ways to say these things as well.

Answer 3

This is not a fully related answer, but it's the best way I know to link polynomials (and formal series) to categorical lingo.

And the answer is analytic functors! :-)

First of all it's easy to regard $\mathbb K$-valued polynomials in $n$ variables as functions $\mathbb{N}^n \to \mathbb{K}$ that are almost always $0$; this in turn means that formal power series $\mathbb{K}[\![ t]\!]$ can be identified with functions $\mathbb{N}^n \to \mathbb{K}$.

Second, you can define a functor $F\colon {\bf Set}\to {\bf Vec}_{\mathbb K}$ to be analytic if $$ F(X) \cong \bigoplus_{n\ge 0} f(n)\otimes X^{\otimes n} $$ where $f\colon {\bf Set}\to {\bf Vec}_{\mathbb K}$ is called a generating species (from the Latin "specio"). This functor plays the role of the coefficient $a_n\in\mathbb K$, which is precisely the value of $a\colon \mathbb{N}\to \mathbb K$. This means that the functor $F$ admits an "expansion in Taylor series".

Of course a multivariable functor $F$ is analytic if it acts on objects as $$ \begin{multline} F(\underline X) = F(X_1,\dots X_n) \cong \bigoplus_{\underline n} f(\underline n)\otimes \underline X^{\underline n} = \\ =\bigoplus_{k\ge 0} \bigoplus_{(n_1,\dots, n_k)\in\mathbb N^k} f(n_1,\dots, n_k) \otimes X_1^{n_1}\otimes\dots\otimes X_k^{n_k} \end{multline} $$ Now, the same property (i.e. some sort of cartesian closure, as mentioned in another answer) ensuring that polynomials can be interpreted as polynomial functions allows you to evaluate polynomial functors on objects.