nLab product law

Idea

The product law or product rule or Leibniz rule of differentiation says that for $f,g : X \to \mathbb{R}$ two differentiable functions and $f g$ their (pointwise) product the derivative of their product is

$$d (f g) = (d f) g + f (d g) \,.$$

Generalized to differential forms the product law says that $d$ is a derivation of degree +1 on the graded commutative algebra of differential forms:

$$d (f \wedge g) = (d f) \wedge g + (-1)^{\deg f} f \wedge (d g) \,$$

if $f$ is homogeneous.

Categorification

One categorified form of the product rule occurs in the theory of species. Let $V$ be a symmetric monoidal category with finite coproducts over which the tensor product distributes, and let $FB$ be the core groupoid consisting of finite sets and bijections. Recall the convolution product for species $F, G: FB \to V$ is given by the formula

$$(F \otimes G)[S] = \sum_{S = T + U} F[T] \otimes G[U]$$

and the derivative of $F$ is given by the formula

$$F'[S] = F[S \sqcup \{\ast\}].$$

Then it is easy to check that the canonical map

$$F' \otimes G + F \otimes G' \to (F \otimes G)'$$

is an isomorphism. More generally, for any finite set $X$ one can define an $X^{th}$ derivative by the formula

$$F^{(X)}[S] = F[S + X]$$

and then one has a canonical isomorphism

$$(F \otimes G)^{(X)} \cong \sum_{X = Y + Z} F^{(Y)} \otimes G^{(Z)}$$

which is a categorification of the general product rule

$$(f \cdot g)^{(n)} = \sum_{n = j + k} \frac{n!}{j! k!} f^{(j)} \cdot g^{(k)}.$$

Related concepts

• derivation

• multiplicative spectral sequence