Draft:Homotopy theory
- This draft page will be used to work out sections on the homotopy groups of spheres and shape theory
Pro-homotopy-type and shape theory
Homotopy theory works the best with spaces with nice local behaviors; e.g., CW complexes or absolute neighborhood retracts. Shape theory extends homotopy theory to spaces with poor local behaviors. A canonical example is a Warsaw circle.
In mathematics, especially homotopy theory, a homotopy quotient is a homotopy-theoretic quotient as opposed to a set-theoretic one. Precisely, this is the homotopy colimit of the action groupoid. Especially in differential geometry, it is also known as the Borel construction.
Quotient groupoid
Let
be an action groupoid (usual set-theoretic). The colimit of the above diagram in Set is the coequalizer of it; i.e., the usual set-theoretic quotient.
Reference
- Jardine, John F. (2015). Local homotopy theory. Springer Monographs in Mathematics. New York: Springer-Verlag. section 9.2. doi:10.1007/978-1-4939-2300-7. ISBN 978-1-4939-2299-4. MR 3309296.
Further reading
- "Homotopy quotient". ncatlab.org.
- "Example 5.3.2.15 (Homotopy Quotients)". Kerodon.
- Anel, Mathieu. "The homotopy quotient When the notations mean more than what we make them say" (PDF).
- "Preview". Equivariant Cohomology in Algebraic Geometry. 2023. pp. 1–8. doi:10.1017/9781009349994.002. ISBN 978-1-009-34999-4.
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