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 $i,j \in I$
Then
In constructive mathematics, it is necessary that $I$ have decidable equality; alternatively, one could let the Kronecker delta take values in the lower reals.
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.
Named after Leopold Kronecker.
See also