Homotopy category

The homotopy category of based CW complexes has spaces as objects and based homotopy classes as morphisms. Composition is well-defined because composition with continuous maps preserves homotopies. A based homotopy equivalence becomes an isomorphism. Rational localization instead inverts rational homotopy equivalences, producing a category represented by rational spaces. More general model categories have analogous homotopy categories obtained by inverting their weak equivalences.

For maps $A\to X$ and $A\to Y$, the double mapping cylinder $X\bigsqcup (A\times I)\bigsqcup Y$, with each endpoint attached by its given map, models the homotopy pushout. A map out of this model is a pair of maps on $X$ and $Y$ together with a homotopy between their restrictions to $A$. This is why compatible homotopy classes extend over a homotopy pushout. For a CW pair, the inclusion is a cofibration and the ordinary pushout gives the same homotopy type.

A based topological space with a continuous map $m$ whose left and right restrictions at the basepoint are homotopic to the identity. Associativity and inverses up to homotopy are additional conditions, not part of the bare definition.

An H-space whose multiplication is homotopy associative and admits a homotopy inverse. These operations give $[X,Y{]}_{\ast }$ a group structure for every based CW complex $X$. A loop space is the basic example: concatenate paths, use reversal for inversion, and use reparametrization for homotopy associativity. A bare H-space should not automatically be treated as a group object.

For a contravariant pointed functor on based CW homotopy types, half-exactness on a cofibration means that $F(X/A)\to F(X)\to F(A)$ is exact at $F(X)$: the image is the inverse image of the distinguished point. This does not demand injectivity or surjectivity at the ends. Set-valued exactness and group-valued exactness must be distinguished.

A reduced contravariant homotopy functor on based CW complexes satisfying the wedge-to-product axiom and the Mayer-Vietoris weak-pullback gluing axiom is represented by $[X,Y{]}_{\ast }$ for a CW complex $Y$. Half-exactness alone is not the full hypothesis. A reduced generalized cohomology theory supplies these axioms; suspension compatibility assembles its representing spaces into an Omega-spectrum.