natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category 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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 $K(G,n)$ 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 an 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 $(HG)_n \equiv K(G,n)$.
Generalized reduced cohomology satisfies the Eilenberg-Steenrod axioms:
( Suspension ) There is a natural isomorphism
( Exactness ) For any cofiber sequence $X \to Y \to Z,$ the sequence
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
is an isomorphism.
Ordinary cohomology also satisfies the dimension axiom:
The notion of Whitehead generalized cohomology, i.e. with coefficients in any spectrum type:
In the further generality of twisted cohomology with coefficients in any spectrum type:
Implementation of ordinary integral cohomology (i.e. with coefficients in the Eilenberg-MacLane spaces/-spectra) in cubical Agda:
Last revised on February 2, 2023 at 14:10:00. See the history of this page for a list of all contributions to it.