Homotopy Type Theory Synthetic homotopy theory > history (Rev #19, changes)

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This page is under construction. - Ali

Open problems
~~Basic~~ Rough ~~notions~~ list
~~Constructions~~ Spaces ~~and~~ ~~concepts~~Homotopy groups
Spaces
Homological Algebra
Fibrations
Misc.
Useful references

Open problems

See open problems

Basic Rough notions list

Higher inductive types

Type families $P : A \to \mathcal{U}$ and fibrations $P(\star_A ) \to \sum_{(a:A)}P(a) \to A$

Feel free to add and reorganise.

~~Constructions and concepts~~

Higher inductive types
Type families $P : A \to \mathcal{U}$ and fibrations $P(\star_A ) \to \sum_{(a:A)}P(a) \to A$
homotopy groups of spheres
pushout $Y \sqcup^X Z$
- join $X * Y$
- suspension $\Sigma X$
- wedge sum $X \vee Y$
- homotopy fiber
- homotopy cofiber?
- smash product $X \wedge Y$
Eilenberg-MacLane spaces $K(G,n)$
James construction $J(X) \simeq \Omega \Sigma X$
localization $L X$
homology?
cohomology $H^n(X;A) \equiv \| X \to K(A,n) \|_0$
spectral sequences
Hopf construction: H-space on $A$ gives fibration $A \to A * A \to \Sigma A$
Whitehead product
loop space of a wedge of circles
spectrum
Gysin sequence?
Hopf invariant?
Mayer-Vietoris sequence?
Blakers-Massey Theorem?
Serre spectral sequence?
Atiyah–Hirzebruch spectral sequence?
Seifert-van Kampen theorem
CW complex?
Cellular cohomology?
covering spaces?

~~Homotopy groups~~

Spaces

~~homotopy~~ circle ~~groups~~ of ~~spheres~~

spheres

interval?

real projective space?

~~Spaces~~

pushout $Y \sqcup^X Z$

join $X * Y$

suspension $\Sigma X$

wedge sum $X \vee Y$

homotopy fiber

homotopy cofiber?

smash product $X \wedge Y$

Eilenberg-MacLane spaces $K(G,n)$

James construction $J(X) \simeq \Omega \Sigma X$

localization $L X$

~~Homological Algebra~~

homology?

cohomology $H^n(X;A) \equiv \| X \to K(A,n) \|_0$

spectral sequences

~~Fibrations~~

~~Hopf construction: H-space on $A$ gives fibration $A \to A * A \to \Sigma A$~~

~~Misc.~~

Whitehead product

loop space of a wedge of circles

I would like to include some articles about spectra here. This would probably absorb homological algeba. Floris van Doorn’s thesis should be a good start.

Ali

Useful references

On the homotopy groups of spheres in homotopy type theory , ~~Guillaume~~ ~~Brunerie’s~~ ~~thesis.~~ ~~Covers:~~Guillaume Brunerie

Sequential colimit?

James construction

Whitehead product

Smash product and it’s monoidal structure

Cohomology rings

Mayer-Vietoris sequence?

Hopf invariant?

Gysin sequence?

Computation of $\pi_4(S^3)=\mathbb{Z}/n\mathbb{Z}$ and later $n=2$ .
On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory , ~~Floris~~ ~~van~~ ~~Doorn’s~~ ~~thesis~~Floris van Doorn

Eilenberg-Maclane spaces

Smash product and it’s monoidal structure

Spectral sequence

Serre spectral sequence

Spectral sequence for Cohomology

Spectral sequence for Homology

Computations of cohomology of spheres

Spectra?

Generalised cohomology theories
Higher-Dimensional Types in the Mechanization of Homotopy Theory , ~~Favonia’s~~ ~~thesis~~Favonia

Covering spaces

Seifert-van Kampen theorem

Blakers-Massey Theorem

CW complexes

Cellular cohomology

HoTT book

category: homotopy theory, navigation

Revision on October 19, 2018 at 10:52:12 by Ali Caglayan. See the history of this page for a list of all contributions to it.