nLab
product law

Contents

Contents

Idea

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

d(fg)=(df)g+f(dg). d (f g) = (d f) g + f (d g) \,.

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

d(fg)=(df)g+(1) degff(dg) d (f \wedge g) = (d f) \wedge g + (-1)^{\deg f} f \wedge (d g) \,

if ff is homogeneous.

Categorification

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

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

and the derivative of FF is given by the formula

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

Then it is easy to check that the canonical map

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

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

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

and then one has a canonical isomorphism

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

which is a categorification of the general product rule

(fg) (n)= n=j+kn!j!k!f (j)g (k).(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. See the history of this page for a list of all contributions to it.