# nLab motivic function

## Ingredients

• cohomology

• stable homotopy theory

• ## Definitions

• pure motive

• mixed motive

• noncommutative motive

• motivic function

• motivic integration

• motivic Donaldson-Thomas invariant

• # Contents

## Definition

Given a small category $\mathcal{C}$ with coproducts and given an object $X \in \mathcal{C}$, the abelian group $Mot(X)$ of motivic functions on $X$ is defined by generators and relations as follows: it is the quotient of the free abelian group on the morphisms $S \to X$ by the relations

$[S_1 \coprod S_2 \stackrel{(f_1,f_2)}{\to} X] = [S_1 \stackrel{f_1}{\to} X] + [S_1 \stackrel{f_2}{\to} X] \,.$

The construction of motivic functions has some similarity with

## References

Section 2.2 of

Last revised on June 13, 2013 at 01:32:29. See the history of this page for a list of all contributions to it.