symmetric monoidal (∞,1)-category of spectra
A factorization algebra is an algebra over an operad where the operad in question is like the little disk operad, but with each disk embedded into a given manifold $X$.
This way a factorization algebra is an assignment of a chain complex $V_D$ to each ball $D \subset X$ embedded in $X$, and for each collection of non-intersecting embedded balls $D_1 , \cdots, D_n \subset D \subset X$ sitting inside a bigger embedded ball $D$ in $X$ a morphism
such that composition of such operations is suitably respected.
For $X$ a topological space write $Fact_X$ be the colored operad in Set whose
objects are the connected open subsets of $X$;
the hom-set $Fact_X(\{U_i\}_i, V)$ is the singleton precisely if the $U_i$ are all in $V$ and are pairwise disjoint and is the empty set otherwise.
This specifies composition uniquely.
For $(C, \otimes)$ a symmetric monoidal abelian category let $End(C)$ be its endomorphism operad. A prefactorization algebra on $X$ with values in $C$ is an algebra over an operad over $Fact_X$ in $C$, hence a morphism of operads
These definitions appear (here).
For $X$ a topological space and $U \subset X$ an open subset, a open cover $\{U_i \hookrightarrow U\}_{i \in I}$ is called a factorizing cover if for every finite set of points $\{x_1, \cdots, x_k\} \subset U$ there is a finite subset $\{U_{i_j}\}_{j \in J \subset I}$ 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 $\{U_i \to U\}_{i \in I}$ write $P I$ for the set of finite subsets $\alpha \subset I$ such that for $j,j' \in \alpha$ we have $U_j \cap U_{j'} = \emptyset$.
Given a prefactorization algebra $\mathcal{F}$ and $\alpha \in P I$ write
and for $\alpha_1, \cdots, \alpha_k \in P I$ write
For each $1 \leq i \leq k$ there is a canonical morphism
A prefactorization algebra $\mathcal{F} : Fact_X \to End(C)$ is called a factorization algebra if for every open subset $U \subset X$ and every factorizing cover $\{U_i \to U\}_{i \in I}$ the sequence
is an exact sequence.
These definitions appear here.
See also at cosheaf.
Let now $(C,\otimes)$ specifically be a category of chain complexes.
A [prefactorization algebra] $\mathcal{F} : Fact_X \to End(X)$ is a homotopy factorization algebra if for all factorizing covers $\{U_i \to U \subset X\}_{i \in I}$ the canonical morpshim
is a quasi-isomorphism, where the differential on the left is defined by (…).
This is the analogue of a descent condition for simplicial presheaves.
These definitions appear here.
Factorization algebras have some similarity with
duality between algebra and geometry in physics:
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
This discusses how locally constant factorization algebras obtained from En-algebras induce extended FQFTs.
A fairly comprehensive account of factorization algebras as a formalization of perturbative quantum field theory (see at factorization algebra of observables) is in
Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory
Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)
and the beginning of
Lecture notes include
Kevin Costello (with Owen Gwilliam), Factorization algebras in perturbative quantum field theory in Strings, Field, Topology, Oberwolfach report No 28, 2009 (pdf)
This can also be found mentioned in the talk notes of the Northwestern TFT Conference 2009, see in particular
notes by Christoph Wockel, Talk by Kevin Costello
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