David Corfield cohomology in mathematics

Case study for the dynamics of mathematical reasonin my extension of Friedman's Dynamics of Reason.

Idea

The first cohomology theory appeared in 1935 in a paper by J. W. Alexander, On the chains of a complex and their duals.

The origins of cohomology theory are found in topology and algebra at the beginning of the last century but since then it has become a tool of nearly every branch of mathematics. It’s a way of life! (Ulrike Tillmann, Cohomology Theories)

Cohomology theory has now spread over the whole of mathematics through differential equations, differential operators and so forth. (Peter Hilton, A Brief, Subjective History of Homology and Homotopy Theory in This Century, Mathematics Magazine, Vol. 61, No. 5 (Dec., 1988), pp. 282-291.)

One may seek unity in mathematics through the eyes of cohomology. (June Huh, Combinatorics and Hodge theory, p. 2)

Cohomology Detects Failures of Classical Mathematics

This page will treat cohomology as an example of ‘empirical’ findings becoming principles in a foundational framework in just the kind of way we would expect a revolution in mathematics of significance to mathematical physics to take place. For those wanting to know more about cohomology, here are some intuitions.

In the last century, especially following the work of Grothendieck, cohomology theories have emerged as extremely important tools in algebraic geometry and number theory: they lie at the heart of some of the deepest theorems and conjectures in both subjects. For example, classical Hodge theory, which studies the singular cohomology with real/complex coefficients for complex varieties, is a central topic in modern algebraic geometry, with applications throughout the subject and beyond. Likewise, $p$-adic Hodge theory, which studies the $p$-adic cohomology of $p$-adic varieties, is an equally fundamental notion in arithmetic geometry: it provides one of the best known tools for understanding Galois representations of the absolute Galois group of $\mathbf{Q}$. Moreover, unlike in the complex setting, there is a large number of cohomology theories in the $p$-adic world: étale, de Rham, Hodge, crystalline, de Rham–Witt, etc. (Bhatt, 2112.12010)

Rapid History

• Rattlebag of homology and cohomology theories, including de Rham cohomology by 1930s.
• Organised by Eilenberg-Steenrod (1952) by means of the language of category theory, Eilenberg-Mac Lane (1945).
• Priority given to cohomology over homology (latter a derived notion)
• Cohomology for groups as well as spaces.
• Extraordinary or generalized cohomologies found.
• Weil and cohomology for number theory
• Hundreds of varieties of cohomology
• Cohomology organised as $\pi_0H(X, A)$ in $(\infty, 1)$-topos by means of higher category theory (duality with homotopy).
• This higher category theoretic framework and the notion of cohomology as just what is needed to understand and advance current physics

Friedmannian account

1920s-1930s

• Constitutive language: algebra and topology, as expressed in naive set theory.
• Theories: various defined homology and cohomology theories, associating algebraic entities to spaces
• Observations: some regularities found, e.g., (simpler) spaces give the same results for any theory, homotopy invariance.

By 1952 (work done in 1940s)

• Constitutive language: category theory
• Theories: axiomatised (co)homology, Eilenberg-Steenrod axioms, includes some previously observed properties as axioms.
• Observations: Cech ‘homology’ no longer a homology. (There is a “corrected” theory known under the name strong homology.)

“lenberg and Steenrod initiated a new approach by focusing not on the machinery used for the construction of homology or cohomology groups, but on the properties shared by the various theories. The selected a small number of these properties and took them as \textit{axioms} for a theory of homology and cohomology; they showed that many other properties, formerly separately proved for each theory, were in fact consequences of the axioms, and they examined each theory accordingly to see if it satisfied the axioms.’‘ (Dieudonné 2009, 11)

Characteristic classes. After the observation of ‘generalized’ or ‘extraordinary’ cohomologies, satisfying all but dimension axiom, e.g., K-theory.

Other axioms discarded but when? Dropping the suspension axiom leads to nonabelian cohomology and dropping the “homotopy axiom” (and taking the domain spaces to be smooth manifolds) leads to the further generality of differential cohomology. :

1959

• Category theory
• The Brown representability theorem for generalized (Eilenberg-Steenrod) cohomology: $H^q_E(X) = lim_n [S^n X; E_{q + n}]$, for a spectrum $E_i$.
• Allows many new cohomologies, e.g., various cobordism theories (Thom), relating to all quarters of mathematics.

Flourishing of cohomology theories, including sheaf cohomology. Understanding of generalized cohomology as (fully) abelian cohomology. Rise of nonabelian cohomology. Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology (1973), establishes that much of cohomology is about Hom sets. Rise of topos theory and then $(\infty, 1)$-toposes.

By the 2010s

• Constitutive language: $(\infty, 1)$-topos theory/univalent type theory/homotopy type theory
• Theories: thousands of examples of cohomology are components of Hom-space in a $(\infty, 1)$-topos; differential cohomology in cohesive $(\infty, 1)$-topos.
• Observations: ?

Eilenberg-Steenrod axioms for homology theories have a modern formulation in terms of ∞-category theory. (4.2.3 of Topo-logie)

Idea of Lurie of function into space, functions out of space, and torsors. If cohomology is Hom space, then is it a question of providing subtle enough coefficient objects?

Formal group laws, universal cohomology theory, differential cohomology.

Slow uptake

Princeton Companion to Mathematics edited by Timothy Gowers, reviews

• Birch: I note that cohomology gets short shrift; it is a valuable and pervasive technique, but may be hard to write about attractively.

• Donaldson: I was hoping to find a broad discussion of the influence of cohomology in various guises (surely one of the main developments of the twentieth century), but was disappointed. It would have been interesting and topical to see more on quantum field theory, as a notable idea “that mathematicians are grappling with at the beginning of the twenty-first century” (although there is some coverage of this under the headings “Mirror symmetry” and “Vertex operator algebras”).

• Macintyre: For an idea so pervasive in modern mathematics, cohomology gets rather little coverage, except for three pages in Totaro’s beautiful paper. One can hope for much more in a revised edition.