[[!redirects Cohomology]] ## 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]]: $$ \bar{H}^n(X ; G) \equiv \| X \to^* K(G,n) \|_0 $$ 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: $$ \bar{H}^n (X; E) \equiv \| X \to E_n \|_0 $$ 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]] category: homotopy theory