nLab Kronecker delta

Definition

For $I$ a set, the Kronecker delta-function is the function $I \times I \to \{0,1\}$ which takes the value 0 everywhere except on the diagonal, where it takes the value 1.

Often one writes for elements $i,j \in I$

$$\delta^{i}_j \coloneqq \delta(i,j) \,.$$

Then

$$\delta^i_j = \left\{ \array{ 1 & if i = j \\ 0 & otherwise } \right.$$

In constructive mathematics, it is necessary that $I$ have decidable equality; alternatively, one could let the Kronecker delta take values in the lower reals.

Generalizations

• The Kronecker delta is the characteristic map of the diagonal function into $I \times I$. More generally, we may call the characteristic morphism of a diagonal morphism in any category with a subobject classifier a “Kronecker delta”.

• In the context of profunctors/graphs of functors, we can view the Kronecker delta as the decategorification of the Hom profunctor.

Related concepts

• Einstein summation convention

• Levi-Civita symbol

• Dirac delta distribution

• graph of a functor

References

Named after Leopold Kronecker.

See also

• Wikipedia, Kronecker delta