---
date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 21:09:38
title: factorization homology
aliases: [factorization homology, topological chiral homology]
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Last modified date: <%+ tp.file.last_modified_date() %>
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- Tags
- #higher-algebra
- Refs:
- Masterclass: #resources/videos
- #resources/papers
- Links:
- [THH](topological%20Hochschild%20homology)
- [scanning map](scanning%20map)
- [nonabelian Poincare duality](nonabelian%20Poincare%20duality.md)
- [Dold-Thom theorem](Dold-Thom%20theorem)
- [cobordism hypothesis](Unsorted/cobordism%20hypothesis.md)
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# factorization homology
Relation to [Hochschild homology](Unsorted/HH.md): $$\HoH(A) = \int_{S^1} A$$
# Motivations
- Motivations: factorization homology forms an important class of [topological field theories](Unsorted/TQFT.md): the ones in which the global [observables](observables) are determined by the local observables. It can be modeled using labeled [configuration spaces](configuration%20spaces); in fact, it originates from configuration space models for [mapping spaces](mapping%20spaces).
- Descriptions of the factorization homology of free [E_n algebra](Unsorted/E_n%20ring%20spectrum.md) and of $E_n\dash$enveloping algebras of [Lie algebras](Lie%20algebras) are known.
- [suspension spectra](suspension%20spectra) provide another class of algebras for which factorization homology is known.
- This follows from [nonabelian Poincare duality](nonabelian%20Poincare%20duality.md): describes factorization homology of $n$-fold [loop spaces](loop%20spaces) in terms of [mapping spaces](mapping%20spaces) or section spaces.
- For $M = S^1$ , factorization homology specializes to [Topological Hochschild homology](Unsorted/HH.md).
- There is an [equivalence of categories](equivalence%20of%20categories.md) $$\modsleft{E_1\dash A} \cong \bimod{A}{A}.$$
- # Definitions
# Examples
