nLab motivic function

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

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

Related concepts

The construction of motivic functions has some similarity with

• the construction of Baum-Douglas geometric cycles

• the construction that Baez-Dolan called de-groupoidification.

References

Section 2.2 of

• Maxim Kontsevich, Yan Soibelman, Motivic Donaldson-Thomas invariants: summary of results (arXiv:0910.4315)