Q:

Size of the category of cohomology theories

I'd like to understand the structure of the functor category Coh whose objects are cohomology functors from a category of Spaces to the category of graded commutative rings GCR. Spaces could be any of the familiar geometric categories: topological spaces, manifolds, algebraic varieties, schemes, etc.
My first question is about just the size of Coh. For any choice of Spaces there are several well known cohomology functors (singular, de Rham, etale, ...) and new ones keep cropping up (syntomic, prismatic) - and they all agree too, on suitable subcategories of Spaces after suitable extensions of scalars, hinting at a fundamental core to it all (e.g., motives) - but I don't know how many more there can be. Is there a systematic way to enumerate or even construct them all? The latter is extremely unlikely because construction of any one we have has been a highly creative and painstaking task, but can we at least know how many in some sense are still out there? Is it even a discrete set, or can we in fact "deform" cohomology theories in families in certain settings?
Same question of size applies to the sets of natural transformations among cohomology theories, i.e., the Hom sets in Coh. Apart from the standard comparison isomorphisms what do we know about other natural transformations, even just for two well known cohomologies, for example, Betti and de Rham?
Sorry if the scope of the question is too broad and I should have made it more manageable by fixing a particular category of spaces and coefficient system for their cohomology. Please feel free to pick a setting that makes for a satisfactory answer.

A:

Welcome to MathOverflow nms! I guess that there are  at least two possible answers to your question:

in algebraic topology, Brown's representability theorem says that cohomology theories are represented by spectra, so you can construct a new cohomology theory by constructing a new spectrum. The category of spectra can also be proved (see Remark 1.4.2.4 in Lurie's "Higher Algebra" for instance) to be presentable, which is a suitable "smallness" condition (see Chapter Five of Lurie's "Higher Topos Theory");
in algebraic geometry, there are also suitable analogues of Brown's representability, which assert that cohomology theories with suitable properties are representable as motivic spectra, by which one means objects in the categories $\mathrm{DA}_\tau(S;\Lambda)$ and $\mathrm{SH}_\tau(S;\Lambda)$ built out of a scheme $S$, a ring $\Lambda$ and a Grothendieck topology $\tau$ using $\mathbb{A}^1$-homotopy theory (see Ayoub's thesis or the book of Cisinski and Déglise for a detailed analysis of the construction of these categories). Such representability results appear for instance in the paper "Mixed Weil cohomologies" by Cisinski and Déglise, or in §1 of the paper "The rigid syntomic ring spectrum" by Déglise and Mazzari, or in Drew's thesis. I suggest also taking a look at the appendix of the book "Many variations of Mahler Measures" by Brunault and Zudilin, for an application of this representability theorems to Deligne-Beilinson cohomology. Finally, questions about the sizes of the sets of morphisms in the categories $\mathrm{DA}_\tau(S;\Lambda)$ and $\mathrm{SH}_\tau(S;\Lambda)$ are very difficult to answer: for instance, it seems completely out of reach to prove that the motivic cohomology groups $\mathrm{Hom}(\mathbf{1}_X,\mathbf{1}_X(j)[i])$ are finitely generated, under suitable assumptions on $X$ (e.g. $X$ smooth and proper over $\mathbb{Q}$).

A:

It is perhaps more natural to study homology theories (or cohomology theories) up to `Bousfield equivalence', where two theories are equivalent if they send the same maps to isomorphisms.  (So, for example, classical cohomology with coefficients in a field $k$ are sorted by the characteristic of $k$.)   This has been studied, with much sophistication, beginning with papers by Bousfield around 1970, who wrote about the lattice of such localization functors, and gave examples of both orderly and unusual behavior.
I would suggest looking up his papers, and also those of Mark Hovey (e.g. Cohomological Bousfield classes. J. Pure Appl. Algebra 103 (1995), no. 1, 45–59, or  Hovey, Mark; Palmieri, John H. The structure of the Bousfield lattice. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 175–196, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.)   A paper that shows that versions of your question can be proved to be undecidable is  Casacuberta, Carles; Scevenels, Dirk; Smith, Jeffrey H. Implications of large-cardinal principles in homotopical localization. Adv. Math. 197 (2005), no. 1, 120–139.  Have fun exploring this!
[I should add that, when restricted to the category of finite CW complexes, the Bousfield classes are known: they are detected by the sequence of Morava K-theories associated to each ordinary prime.]

