under construction
Madhav Nori invented an original approach to mixed motives, in the spirit of Tannakian theory, but based on a specific Nori's Tannakian theorem. Unfortunately his original work, which has been lectured on some summer schools and conferences is not widely available in written form. Maxim Kontsevich based his approach to periods and their appearance in deformation quantization partly on Nori’s insight.
The construction of Nori motives themselves has been generalised to categories over a base S by Arapura and Ivorra. Arapura’s approach is based on constructible sheaves. His categories allow pull-back and push- forward, the latter being a deep result. The same paper also constructs the weight filtration on Nori motives and establishes the equivalence between Nori motives and André pure motives. Ivorra’s approach is based on perverse sheaves. Compatibility under the six functors formalism? is open in his setting [Huber-Stach]
It is known that Nori’s and Ayoub’s Motivic galois groups agree.
The subcategory of 1-motives in Nori motives agrees with Deligne’s 1-motives, and hence also with 1-motives in Voevodsky’s category.
Annette Huber, Stefan Müller-Stach, with contributions of Benjamin Friedrich and Jonas von Wangenheim, Periods and Nori motives, (Book in preparation: first version, September 2015, 323 p. )
Annette Huber, Stefan Müller-Stach, On the relation between Nori motives and Kontsevich periods, 1105.0865
Madhav Nori, TIFR notes on motives, unpublished, pdf
MathOverflow: kontsevichs-conjectures-on-the-grothendieck-teichmuller-group
Alain Bruguieres, On a tannakian result due to Nori, dvi, ps, pdf, djvu
Jonas von Wangenheim, Nori-Motive und Tannaka-Theorie, arxiv/1111.5146
Isamu Iwanari, Tannakization in derived algebraic geometry, arxiv/1112.1761
Joseph Ayoub, Luca Barbieri-Viale, Nori 1-motives (arXiv:1206.5923)
Utsav Choudhury?, Martin Gallauer de souza?, An isomorphism of Motivic galois groups (arXiv:1410.6104)
Joseph Ayoub, Luca Barbieri-Viale, Nori 1-motives, pdf
Nori motives over a base S
Florian Ivorra, Perverse Nori motives (hal-00978893)
Donu Arapura, An Abelian category of Motivic sheaves (arXiv:0801.0261)
Characterization of the universality of Nori motives via Barr completion of the syntactic category of the theory (in the formal logical sense) of singular cohomology of schemes is in
For more on this topos theory perspective, see
Luca Barbieri-Viale, T-Motives (arXiv:1602.05053)
Olivia Caramello, Motivic toposes (arXiv:1507.06271)