David Corfield cohomology

Case study for my extension of Friedman's Dynamics of Reason.

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. P. Hilton, A Brief, Subjective History of Homology and Homotopy Theory in This Century, Mathematics Magazine, Vol. 61, No. 5 (Dec., 1988), pp. 282-291.

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
• 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.)

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

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

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.

Last revised on January 23, 2022 at 10:56:30. See the history of this page for a list of all contributions to it.