Note that is coexact, since We can iterate this construction, forming the mapping cone of mapping cone, etc. Note that , so the LES is just forming iterated suspension.
Cofibrations are good embeddings and satisfies the homotopy extension property. The prototypical examples are inclusion to mapping cylinder for . Equivalently, has HEP iff is a retract of . Since (since is compact the Hom-tensor adjunction works fine), it is the same as the following diagram.
Fibrations are like fiber bundles and satisfies the homotopy lifting property (more precisely it is called Hurewicz fibrations). The prototypical examples are the path space fibration and pullback along any continous map . A Serre fibration is just having the homotopy lifting proprety with respect to .
We can show that a Serre fibration has the HLP for all CW complexes by induction on skeletons using the fact that is homeomorphic to the pair .
If is a cofibration, then is a Serre fibration (use the tensor-hom adjunction). Corollary: is a fibration.
- Essentially the theory of model category gives something like factorization system (surjectives follow by injectives) but slightly weaker (not requiring the factorization to be functorial), i.e. the notion of weak factorization system. Via it we can formulate the notion of a model category succinctly.
We can check that has HEP, hence so is by closure under pushout. We can use it to show that is a homotopy equivalence by proving a more general lemma that if is contractible and has HEP, then is a homotopy equivalence.
Where does group structure of higher homotopy group come from? We can check is a cogroup object in the homotopy category (co--group), and that if is any pointed space and is any co--group, so is (the smash product). This is similar to the fact that is a group for any set and group .
Recall the Eckmann-Hilton argument: If is a co--group in two ways, say and , and is a cohomomorphism for the -structure (and vice versa) then and is cocommutative. A corollary is that is a cocommutative group. The idea behind the Eckmann-Hilton argument is the very simple observation that group objects in the category of groups are precisely abelian groups. For more on Eckmann-Hilton argument, see this.
We see that the homotopy category is very closed to being a triangulated category, except that is not generally invertible (e.g. see this for an example). Inverting gives the category of spectra which is triangulated.
Since has a -coaction, i.e. there is a map by crushing the middle circle of the cone. As a consequence, is a -set, and the map is a map as -sets.
Construction of relative homotopy sequence: Starting from Define and , and note that . Alternatively, we can start with the puppe sequence (using mapping fiber rather than mapping cofiber)
One can show by perturbation argument. By the path lifting property, is bijective for . In particular, the higher homotopy groups of all orientable surface vanish.
Interpretation of : If represents zero in , then where . (Proof is by interpreting the homotopy as a homotopy from inverted can to .) Corollary: If is a CW pair and has an -cell, then implies any map is homotopic rel to a map (The advantage of working relatively is that we can proceed to build the homotopy inductively, and it passes to limit). Corollary of corollary: If is an inclusion of subcomplex (both path-connected), and for all , then admits a deformation retract (relative to ). Final corollary: (Whitehead theorem) CW complexes is a homotopy equivalence iff is an isomorphism for all . We use CW approximation to replace by a homotopic map that is a cellular map. This gurantees that we can put a CW structure on such that is an inclusion of CW subcomplex, and it reduces to showing is a homotopy equivalence. The relative homotopy long exact sequence shows that is an isomorphism for all , hence we are done. In particular, if a CW complex has homotopy groups all zero, then it is contractible. This shows that CW complexes are very special objects.
A remark is that an analogous therem holds for chain complexes, if we have chain complexes of projectives, then is a chain homotopy equivalence iff is a quasi-isomorphism (This has to do with the fact that there is a model category structure whose class of cofibrations are maps that are monomorphisms in each degree with projective cokernel).
It is a fact from point set topology that on a paracompact base, a local Serre fibration is a Serre fibration. As a corollary, fiber bundles provide a rich source of Serre fibrations. If is a surjective fibration, then for , implies the map is an isomorphism for all (in other words, the fibration assumption implies that mapping fiber has the same homotopy type as the actual fiber). This implies the homotopy sequence of a fibration (in particular recover result for covering space).
The fundamental group of the base acts on every algebraic invariant attached to the homotopy type of the fiber. There is another description of the monodromy action. Let represented by and is a path from to , then we can just imagine growing radially by .
Recall for . This is actually an equivariant isomorphism. Fact: . This implies that (action by is by the natural shift).
Reference:
motivation for higher topos theory: https://mathoverflow.net/questions/433554/higher-topos-theory-whats-the-moral
How to think about model cateogory: https://mathoverflow.net/questions/287091/why-do-we-need-model-categories