T-motives

Abstract

Considering a (co)homology theory $\mathbb{T}$ on a base category $\mathcal{C}$ as a fragment of a first-order logical theory we here construct an abelian category $\mathcal{A}[\mathbb{T}]$ which is universal with respect to models of $\mathbb{T}$ in abelian categories. Under mild conditions on the base category $\mathcal{C}$, e.g. for the category of algebraic schemes, we get a functor from $\mathcal{C}$ to $\mathrm{Ch}(\mathrm{Ind}(\mathcal{A}[\mathbb{T}]))$ the category of chain complexes of ind-objects of $\mathcal{A}[\mathbb{T}]$. This functor lifts Nori's motivic functor for algebraic schemes defined over a subfield of the complex numbers. Furthermore, we construct a triangulated functor from $D(\mathrm{Ind}(\mathcal{A}[\mathbb{T}]))$ to Voevodsky's motivic complexes.

Added reference to arXiv:1604.00153 [math.AG]

References in corpus (1)

• Nori 1-motives

Cited by in corpus (4)

• Tensor structure for Nori motives
• Tensor product of motives via Künneth formula
• On topological motives
• Universal cohomology theories