Homotopy Type Theory Synthetic homotopy theory > history (Rev #4, changes)

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Open problems

See open problems

Useful constructions

• Homotopy pushout? $Y \sqcup^X Z$
  • Join? $X * Y$
  • Suspension $\Sigma X$
  • Homotopy fiber?
  • Smash product $X \wedge Y$
• Eilenberg Maclane spaces? $K(G,n)$
• James construction $J(X) \simeq \Omega \Sigma X$
• Hopf construction H-space on $A$ gives fibration $A \to A * A \to \Sigma A$
• Localization
• Whitehead product
• Homology?
• Cohomology $H^n(X;A)\equiv \Vert X\to K(\mathrm{<span><del\; class="diffmod">\; G</del><ins\; class="diffmod">\; A</ins></span>},n){\Vert }_0$ H^n(X;A) \equiv \| X \to K(G,n) K(A,n) \|_0
• loop space of a wedge of circles

Useful references

• On the homotopy groups of spheres in homotopy type theory