Homotopy Type Theory cohomology > history (Rev #7, changes)

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Cohomology groups? are algebraic invariants of types?. Often, 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.

Ordinary cohomology denotes cohomology groups with coefficients in \mathbb{Z} this is usually difficult to compute for most spaces, so they are usually broken up into groups for each prime pp with coefficients in p\mathbb{Z}_p. These can be glued back together via the universal coefficient 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 a 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


category: homotopy theory

Revision on June 9, 2022 at 05:09:43 by Anonymous?. See the history of this page for a list of all contributions to it.