nLab ordinary cohomology in homotopy type theory



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




The ordinary cohomology groups are algebraic invariants of homotopy types, and hence of types in homotopy type theory. Typically they are much easier to compute than homotopy groups. There are many theorems in classical algebraic topology relating them other invariants such as the universal coefficient /theorem and the Hurewicz theorem.


There are many different flavours of cohomology, but it’s usually best to start simple and add features according to its use.

Let K(G,n)K(G,n) be the Eilenberg-MacLane space of an abelian group GG for some n:n : \mathbb{N}. The (reduced) ordinary cohomology group (of degree nn with coefficients in GG) of a pointed space XX is the following set:

H¯ n(X;G)X *K(G,n) 0 \bar{H}^n(X ; G) \equiv \| X \to^* K(G,n) \|_0

Note that there is an H-space structure on K(G,n)K(G,n) naturally, so for any |f|,|g|:H n(X;G)|f|,|g| : H^n(X;G) we can construct an element |λx.μ(f(x),g(x))|:H n(X;G)|\lambda x . \mu(f(x),g(x))| : H^n(X; G), hence we have a group.

Note for any type XX we can make this the unreduced cohomology (and call it HH instead of H¯\bar{H}) by simply adding a disjoint basepoint to XX giving us X +X+1X_+ \equiv X + 1 making it pointed.

Let EE be a spectrum, we can define the (reduced) generalized cohomology group of degree nn of a pointed space XX is defined as:

H¯ n(X;E)XE n 0 \bar{H}^n (X; E) \equiv \| X \to E_n \|_0

note that E nE_n has a natural H-space structure as by definition we have E nΩE n+1E_n \simeq \Omega E_{n+1} giving us the same group operation as before. In fact, ordinary cohomology becomes a special case of generalized cohomology just by taking coefficients in the Eilenberg-MacLane spectrum HGHG with (HG) nK(G,n)(HG)_n \equiv K(G,n).


Generalized reduced cohomology satisfies the Eilenberg-Steenrod axioms:

  • ( Suspension ) There is a natural isomorphism

    H¯ n+1(ΣX;E)H¯ n(X;E). \bar{H}^{n+1} (\Sigma X; E) \simeq \bar{H}^{n} (X; E).
  • ( Exactness ) For any cofiber sequence XYZ,X \to Y \to Z, the sequence

    H¯ n(X;E)H¯ n(Y;E)H¯ n(Z;E)\bar{H}^{n} (X; E) \to \bar{H}^{n} (Y; E) \to \bar{H}^{n} (Z; E)

    is an exact sequence of abelian groups.

  • ( Additivity ) Given an indexing type II satisfying 00-choice (e.g. a finite set) and a family X:IU,X: I \to U, the canonical homomorphism

    H¯ n( i:IX i;E) i:IH¯ n(X i;E)\bar{H}^{n} (\bigvee_{i:I} X_i; E) \to \prod_{i:I}\bar{H}^{n} (X_i; E)

    is an isomorphism.

Ordinary cohomology also satisfies the dimension axiom:

  • H¯ n(X,G)=0\bar{H}^{n} (X, G) = 0 if n0.n \neq 0.

See also


See also:

Last revised on June 15, 2022 at 13:04:24. See the history of this page for a list of all contributions to it.