nLab algebra of functions

Definition

Algebra of function on a set

For $R$ a ring and $S$ a set, the set of functions $S \to R$ (to the underlying set of $R$) is itself naturally an associative algebra over $R$, where addition and multiplication is given pointwise in $S$ by addition and multiplication in $R$: for $f_1, f_2 \colon S \to R$ their sum is the function

$$(f_1 + f_2) \colon s \mapsto f_1(s) + f_2(s) \in R \,,$$

their product is the function

$$(f_1 \cdot f_2) \colon s \mapsto f_1(s) f_2(s) \in R$$

and the ring inclusion $R \to [S,R]$ is given by sending $r \in R$ to the constant function with value $r$.

Algebra of functions on an $\infty$-stack

More generally, in the context of (∞,1)-topos theory and higher algebra, there is a notion of function algebras on ∞-stacks.

Properties

Relation to free modules

If $S$ is a finite set or else if one restricts to functions that are non-vanishing only for finitely many elements in $S$, then the algebra of functions with values in $R$ also forms the free module over $R$ generated by $S$.

Hadamard product

If $S$ is a set and $R$ is a commutative ring, then the pointwise multiplication on the function algebra $S \to R$ is the Hadamard product on the function algebra.

Duality between algebra and geometry

Sending spaces to their suitable algebras of functions constitutes a basic duality operation that relates geometry and algebra. For more on this see at Isbell duality.

Related concepts

• pullback of functions
• composition ring