Homotopy Type Theory cohomology > history (Rev #6)

Idea

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 $p$ with coefficients in $\mathbb{Z}_p$. These can be glued back together via the universal coefficient theorem.

Definition

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

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

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

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

Let $E$ be a spectrum, we can define the (reduced) generalized cohomology group of degree $n$ of a pointed space $X$ is defined as:

note that $E_n$ has a natural H-space structure as by definition we have $E_n \simeq \Omega E_{n+1}$ 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? $HG$ with $(HG)_n \equiv K(G,n)$.

Properties

Generalized reduced cohomology satisfies the Eilenberg-Steenrod axioms:

• ( Suspension ) There is a natural isomorphism

  $$\bar{H}^{n+1} (\Sigma X; E) \simeq \bar{H}^{n} (X; E).$$

• ( Exactness ) For any cofiber sequence $X \to Y \to Z,$ the sequence

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

  is an exact sequence of abelian groups.

• ( Additivity ) Given an indexing type $I$ satisfying $0$-choice (e.g. a finite set) and a family $X: I \to U,$ the canonical homomorphism

  $$\bar{H}^{n} (\bigvee_{i:I} X_i; E) \to \prod_{i:I}\bar{H}^{n} (X_i; E)$$

  is an isomorphism.

Ordinary cohomology also satisfies the dimension axiom:

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

See also

• Synthetic homotopy theory
• spectral sequence
• homology?

References

• On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory
• Synthetic Cohomology in Homotopy Type Theory
• HoTT book