In this post I will try to describe how motivic stable homotopy theory bla bla.
Heuristic Sardina picture, made as precise as possible. Classes here are (from the bottom): Weil CTs, Beilinson CTs, Rational coefficient theories and Bloch-Ogus theories, Ordinary theories and oriented theories, All representable theories. Here we write Ordinary theories when we mean ordinary representable theories, and similarly for rational coefficient theories.
Four bigraded theories associated to a spectrum: Overview over how these correspond for example to -theory etc. Also describe what kind of functoriality these typically satisfy. Compare for example Illusie section 4.3 in Motives 1 about rigid cohomology.
Overview of representability theorems, reference to later posts on Weil, on B-O and Beilinson, and on Oriented theories, say whatever there is to say about the remaining classes. Also ref to later posts about Algebraic K-theory, and a later post on algebraic L-theory/Witt groups/hermitian K-theory.
See to what extent we can recover standard functorialities of the four theories. See Gillet in Kth handbook for some examples of functorialities for Chow groups and K-theory (flat maps, proper maps, etc).
Describe non-representable theories with examples. Arithmetic Chow groups, the general problem of inverting A1 in arithmetic.
To understand nature of bigrading, compare the motivic spectral seq with Atiyah-Hirzebruch.
These might be relevant: Etale cohomology of simplicial schemes, Algebraic K-theory, Motivic cohomology, Motivic cobordism, Algebraic cobordism, Etale cobordism, Hermitian K-theory, Witt groups, Continuous etale cohomology, Motivic cobordism
Also others? See Bloch-Ogus theories and Weil cohomology theories, and oriented theories, and several other RG headings on D.
Motivic stable homotopy groups
What about chromatic theory? There are CT pages on algebraic elliptic cohomology and on algebraic Morava K-theory, but are these really defined and known to be representable??? In any case they should be treated here, under this heading.
nLab page on D20 Representable cohomology theories