nLab
differential

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

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

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Basic definitions

Stable homotopy theory notions

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diagram chasing

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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:CCT : C \to C a differential is a morphism d V:VT(V)d_V : V \to T(V) for some object VV such that

Vd VTVT(d V)TTV 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)(-1) morphism that squares to 00.

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 Ω (X)\Omega^\bullet(X) of a smooth manifold XX, which is given by actual differentiation of smooth functions and differential forms. See also differential calculus.

Revised on August 7, 2016 08:52:49 by Zoran Škoda (94.250.137.101)