nLab
differential object

Contents

Definition

A differential object in a category with translation T:CCT : C \to C is an object VV equipped with a morphism d V:VTVd_V : V \to T V.

Remarks

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

Further constructions

  • Usually, when addressing coalgebras for TT as differential objects one considers these in additive categories and requires that they are nilpotent in that Vd VTVTd VTTVV \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:VTVd_V : V \to T V in an additive category CC the shifted differential object is TVT V with differential given by d TV=T(d V) 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)