#
nLab
product law

Contents
# Contents

## 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

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

Last revised on August 16, 2016 at 17:12:17.
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