symmetric monoidal (∞,1)-category of spectra
FQFT and cohomology
This way a factorization algebra is an assignment of a chain complex to each ball embedded in , and for each collection of non-intersecting embedded balls sitting inside a bigger embedded ball in a morphism
such that composition of such operations is suitably respected.
This specifies composition uniquely.
These definitions appear (here).
For a topological space and an open subset, a open cover is called a factorizing cover if for every finite set of points there is a finite subset of pairwise disjoint open subsets such that each point is contained in their union.
Every Hausdorff space admits a factorizing cover.
For a factorizing cover write for the set of finite subsets such that for we have .
Given a prefactorization algebra and write
and for write
For each there is a canonical morphism
A prefactorization algebra is called a factorization algebra if for every open subset and every factorizing cover the sequence
is an exact sequence.
These definitions appear here.
See also at cosheaf.
Let now specifically be a category of chain complexes.
A [prefactorization algebra] is a homotopy factorization algebra if for all factorizing covers the canonical morpshim
These definitions appear here.
Factorization algebras have some similarity with
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
The notion of factorization algebra may be regarded as a slight variation on the concept chiral algebra originally introduced in
A definition formulated genuinely in Higher Algebra appears in section 4.1 Topological Chiral Homology of
and the beginning of
Lecture notes include
This can also be found mentioned in the talk notes of the Northwestern TFT Conference 2009, see in particular
notes by Evan Jenkins on the same talk: Factorization algebras in perturbative quantum gravity
There seems to be a close relation between the description of quantum field theory by factorization algebras and the proposal presented in
The relation of locally constant factorization algebras to higher order Hochschild homology is in
An (infinity,1)-category theoretic treatment of higher factorization algebras is in
A higher-dimensional generalization of vertex algebras is suggested in the framework of factorization algebras in
We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an 'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.