nLab differential object

Definition

A differential object in a category with translation $T : C \to C$ is an object $V$ equipped with a morphism $d_V : V \to T V$.

Remarks

• This says that a differential object is a coalgebra for the endofunctor $T$.

Further constructions

• Usually, when addressing coalgebras for $T$ as differential objects one considers these in additive categories and requires that they are nilpotent in that $V \stackrel{d_V }{\to} T V \stackrel{T d_V}{\to} T T V$ is the zero morphism. Such a differential object is called a chain complex.

• In a differential object $d_V : V \to T V$ in an additive category $C$ the shifted differential object is $T V$ with differential given by $d_{T V} = - T(d_V)$. The minus sign here is crucial in many constructions such as that of the mapping cone. It is naturally understood in terms of fiber sequences in stable infinity-categories.