nLab differential

maybe you are looking for the (total) derivative (differential) of a map

Definition

Abstractly, a differential is the morphism of a differential object defining a chain complex: the boundary operator.

So in a category with translation $T : C \to C$ a differential is a morphism $d_V : V \to T(V)$ for some object $V$ such that

$$V \stackrel{d_V}{\to} T V \stackrel{T(d_V)}{\to} T T V$$

is the zero morphism.

Unwrapping this for the case where the category with translation is a category of chain complexes, it reproduces the ordinary notion of a differential as a degree-$(-1)$ morphism that squares to $0$.

More concretely, the boundary operator on a chain complex is called a differential if this is part of the structure of a dg-algebra on the complex.

Examples

The archetypical example that gives the concept its name is the differential in the de Rham complex $\Omega^\bullet(X)$ of a smooth manifold $X$, which is given by actual differentiation of smooth functions and differential forms. See also differential calculus.