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.

Revised on March 17, 2011 10:41:40
by Urs Schreiber
(77.165.72.45)