T-motives

TL;DR

This paper constructs a universal abelian category for a given (co)homology theory viewed as a logical fragment, and relates it to Nori's and Voevodsky's motivic frameworks for algebraic schemes.

Contribution

It introduces a universal abelian category for (co)homology theories and connects it to established motivic functors and complexes.

Findings

Constructs an abelian category $ ext{A}[ ext{T}]$ universal for models of $ ext{T}$.

Defines a functor from algebraic schemes to chain complexes of ind-objects of $ ext{A}[ ext{T}]$.

Establishes a link between the constructed category and Voevodsky's motivic complexes.

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 ${\rm Ch}({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ 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 $D({\rm Ind}(\mathcal{A}[\mathbb{T}]))$ to Voevodsky's motivic complexes.