Derived, Higher, and Factorization Frameworks: a Boundary Map

“Derived,” “higher-categorical,” and “factorization” name different structural axes. They may occur together, but they are not synonyms and none of them implies either of the others. A derived or homotopical claim must name its weak equivalences and ambient localization or enhancement. A higher-categorical claim must name genuine higher morphisms and their coherence. A factorization claim must name operations for disjoint opens and the applicable Weiss local-to-global condition.

This page is a routing reference: it tells you which label the supplied data licenses and where specialist theory begins. It does not construct a derived QFT, a higher category, or a factorization algebra, and it does not assert that every QFT admits one of these formulations.

Required background. Monoidal, Rigid, and Braided Language separates tensor structure from ordinary composition and from physical tensor factorization; Chain Homotopy, Quasi-Isomorphisms, and Derived Vocabulary separates chain maps, homotopies, quasi-isomorphisms, and localization; Presheaves, Sheaves, Cosheaves, and Čech Descent separates variance, ordinary covers, Čech diagrams, and strict from homotopical descent.

Structure card · Comparison matrix · Derived · Higher · Factorization · QFT example · Triage · Check your understanding

Derived, higher, and factorization structures

Fix the indexing category or site first. A site is a category equipped with specified classes of covers. The index objects might be opens of one manifold, regions of one spacetime, a category of spacetimes, bordisms, or something else. Then record the following fields before applying any framework label.

Objects: what is assigned to each index object?
Maps: which typed morphisms are allowed, and what do they preserve?
Weak equivalences: which class $W$ is to be treated as equivalences?
Higher data: are there $2$ -morphisms, mapping spaces, or coherent higher cells, and at what categorical level?
Monoidal data: is there a tensor product, a disjoint-union operation, or a multi-input structure map?
Variance: do inclusions act covariantly or contravariantly?
Descent: which covers are allowed, and is the comparison a strict isomorphism, a quasi-isomorphism, or another target-specific equivalence?
Physical structure: which topology, causal support, states, positivity, operator domains, or other data enter the conclusion?

A framework may fill only some fields. Missing entries are not supplied by terminology.

Comparison matrix

The rows below overlap. For example, a cochain-valued factorization algebra can occupy the differential graded (dg), derived, and factorization rows at once. The two tables keep independent axes in separate columns.

Objects, weak equivalences, and higher cells

Signal	Objects and ordinary maps	Weak equivalences	Higher cells
Ordinary functor	Objects and morphisms in named categories	Not part of the bare data; categorical isomorphisms remain isomorphisms	None
Complex or dg presentation	Complexes and degree-zero cochain maps; perhaps dg algebras or dg categories	Quasi-isomorphisms are available as maps but are not automatically inverted	Homotopies or mapping complexes only if retained
Derived or localized framework	A relative, model, dg, or stable presentation	A declared class $W$ is inverted	Derived mapping objects only in a stated enhancement
Higher category	$0$ - and $1$ -cells in a specified categorical model	The model’s stated equivalences; localization is optional	Explicit higher morphisms and coherent composition
Ordinary or homotopy (co)sheaf	Values on opens with restriction or assembly maps	Isomorphisms strictly, or a declared target equivalence homotopically	Only if the target or presentation supplies them
Prefactorization algebra	Values on opens and structure-preserving transformations	Target-dependent	Only if the target or presentation supplies them
Factorization algebra	Prefactorization data	Target-dependent, commonly quasi-isomorphisms for cochain complexes	Only if the target or presentation supplies them

Operations, variance, and descent

Signal	Monoidal or disjoint operations	Variance	Class of covers and descent
Ordinary functor	None unless a monoidal structure is supplied	Fixed by the functor’s source	None
Complex or dg presentation	Tensor complexes only in a named dg monoidal target, with Koszul signs	Application-dependent	None
Derived or localized framework	Derived tensor products require replacements and hypotheses	Application-dependent	Derived (co)limits require their own construction
Higher category	Higher-monoidal structure is optional	Application-dependent	Higher descent is optional
Ordinary or homotopy (co)sheaf	No disjoint product is automatic	Contravariant for sheaves; covariant for cosheaves	Strict (co)limit or homotopy (co)limit for a named class of covers
Prefactorization algebra	Compatible multi-input maps for disjoint opens	Covariant under inclusions	No descent axiom yet
Factorization algebra	Disjoint multiplicativity in this convention	Covariant under inclusions	Weiss descent, strict or homotopical as appropriate

An $n$ -ary structure operation is not an $n$ -morphism. Likewise, a homotopy written between two maps does not by itself supply a coherent higher category.

The derived axis: declare what becomes invertible

A useful minimum is a relative category $(\mathcal C,W)$ : a category $\mathcal C$ together with a specified class $W$ of weak equivalences. The word “derived” becomes informative only after the ambient presentation and the treatment of $W$ are stated. Depending on the problem, that presentation may be an ordinary localization, a model category, a dg category, or a stable $\infty$ -category.

For complexes of $R$ -modules, the classical chain of categories is

\operatorname{Ch}(R) \longrightarrow K(R) \longrightarrow D(R)=K(R)[\mathrm{Qis}^{-1}].

The objects can be represented by the same complexes, but the morphisms change. Chain-homotopic maps become equal in $K(R)$ ; quasi-isomorphisms become invertible in $D(R)$ . The finite map from the prerequisite page,

q: \left[ 0\longrightarrow\mathbb Z \xrightarrow{\,2\,}\mathbb Z \longrightarrow0 \right] \longrightarrow (\mathbb Z/2\mathbb Z)[0],

where the two copies of $\mathbb Z$ occupy chain degrees $1$ and $0$ , $q_1=0$ , and $q_0$ is reduction modulo $2$ . This is a quasi-isomorphism without a chain-homotopy inverse. It is therefore invertible in $D(\mathbb Z)$ but not in $K(\mathbb Z)$ . Its formal inverse in the localization is not a missing chain map in the original category.

A differential alone licenses “complex-valued” or “dg,” not “derived.” Nor does derived mean “replace the complex by its cohomology.” An ordinary derived category is a $1$ -categorical localization. Passing to its nerve does not recover mapping complexes or mapping spaces discarded in forming that localization; a dg or stable- $\infty$ enhancement contains additional data. For the classical localization, see Stacks Project Authors 2026, §13.11, tag 05RR, especially Definition 13.11.3. Stable and derived $\infty$ -categorical enhancements are developed in Lurie 2017, Example 1.1.1.12 and §§1.3.2 and 1.3.5, PDF.

The higher-categorical axis: retain morphisms between morphisms

The minimal pattern is

X,Y, \qquad f,g:X\longrightarrow Y, \qquad \alpha:f\Rightarrow g.

Here $\alpha$ is a $2$ -morphism, not the equation $f=g$ . A framework must say whether it is a strict $2$ -category, a bicategory, an $(\infty,1)$ -category, an $(\infty,n)$ -category, or another model, and it must state the relevant notion of equivalence. In an $(\infty,1)$ -category, all morphisms above degree one are invertible in the higher sense; in an $(\infty,n)$ -category, noninvertible morphisms may occur through degree $n$ .

“Higher” does not mean “complicated,” “infinite-dimensional,” or “infinitely many objects.” It means that higher morphism levels and their compositions are retained as structure. The strict $2$ -category $\mathbf{Cat}$ has categories, functors, and natural transformations; it is higher, while this description supplies no weak-equivalence localization or derived construction. Conversely, the ordinary category $D(R)$ is derived without retaining the full mapping data of a dg or stable $\infty$ enhancement.

Changing a strict equation to an isolated homotopy is also insufficient. If a comparison commutes only up to homotopy, compositions of such comparisons need compatible homotopies between homotopies, continuing to the level required by the chosen model. Strictification theorems can remove some coherence in specific model-categorical settings; they do not automatically preserve topology, states, positivity, domains, or every physical structure. The distinction between higher morphisms, coherence, models, and equivalence is treated in Riehl and Verity 2022, preface, pp. ix–xi, and Definitions 1.1.2 and 1.1.10.

The factorization axis: disjoint operations plus Weiss descent

Let $M$ be a manifold and let $\mathcal C^\otimes$ be a specified symmetric monoidal target. A prefactorization algebra assigns $\mathcal F(U)\in\mathcal C$ to each open $U\subseteq M$ and, for pairwise disjoint opens $U_1,\ldots,U_n\subseteq V$ , supplies

\mu_{U_1,\ldots,U_n;V}: \mathcal F(U_1)\otimes\cdots\otimes\mathcal F(U_n) \longrightarrow \mathcal F(V). \tag{1}

These maps must be compatible with permutations, inclusions, and iterated operations. In the nonunital convention, the nullary operation is omitted. For disjoint $U,V$ , the convention used here also asks the canonical map

\mathcal F(U)\otimes\mathcal F(V) \longrightarrow \mathcal F(U\sqcup V) \tag{2}

to be an equivalence for a factorization algebra. This multiplicativity condition is distinct from descent.

A Weiss cover $\{U_i\}_{i\in I}$ of an open $U$ requires every finite subset of $U$ to lie in some $U_i$ . A factorization algebra additionally satisfies the associated local-to-global comparison. For cochain complexes, the homotopical form is schematically

\operatorname*{hocolim}_{[p]\in\Delta^{\mathrm{op}}} \left( \bigoplus_{i_0,\ldots,i_p} \mathcal F(U_{i_0}\cap\cdots\cap U_{i_p}) \right) \longrightarrow \mathcal F(U), \tag{3}

and the comparison is required to be a quasi-isomorphism. The full simplicial Čech diagram matters. In a different homotopical target, “equivalence” in (3) must be replaced by that target’s declared notion; an ordinary coequalizer is not automatically enough. These prefactorization operations, Weiss covers, multiplicativity, and strict versus derived descent are distinguished in Costello and Gwilliam 2023, Definitions 1–3 and Remarks 1–2, pp. 2–3 and 10–12.

Thus a cosheaf need not carry disjoint multiplication, and a prefactorization algebra need not satisfy descent. An arbitrary covariant assignment on opens is neither one. A locally constant factorization algebra on $\mathbb R^n$ can be compared with an $E_n$ -algebra under appropriate target, homotopical, and tangential hypotheses. That is a theorem, not the definition of a factorization algebra, and a general factorization algebra need not be locally constant. The scoped locally constant comparison appears in Lurie 2017, §5.4.5, PDF.

Prefactorization does not force descent

Let $k$ be a field and define a nonunital vector-space-valued assignment on $\mathbb R$ by

\mathcal P(U)= \begin{cases} k,&U=\mathbb R,\\ 0,&U\subsetneq\mathbb R. \end{cases}

Use the identity for the unary operation on $\mathbb R$ and the unique zero map for every other allowed operation. All unit-free permutation and composition diagrams commute, so this is a deliberately degenerate prefactorization algebra.

The collection of all bounded open intervals is a Weiss cover of $\mathbb R$ : every finite subset lies in one sufficiently large interval. Every finite intersection appearing in its Čech diagram is a proper open and has value $0$ . Hence the homotopy colimit on the left of (3) is $0$ , while $\mathcal P(\mathbb R)=k$ . The descent map cannot be an equivalence. This gives a direct counterexample to

\text{prefactorization} \Longrightarrow \text{factorization}.

A controlled QFT-facing classification

Suppose a source proposes observables for a massive free scalar field on the opens of one fixed spacetime. For each $U$ it gives a cochain complex

(\operatorname{Obs}(U),Q_U),

and for each inclusion $j:U\hookrightarrow V$ a degree-zero cochain map

j_*:\operatorname{Obs}(U)\longrightarrow\operatorname{Obs}(V), \qquad Q_Vj_*=j_*Q_U. \tag{4}

Identity and composition make this a dg precosheaf. The differential does not yet make the assignment derived, higher-categorical, prefactorization, or factorization data.

Now suppose the source declares objectwise quasi-isomorphisms to be weak equivalences and works in a stated localization or homotopical enhancement. That supplies the derived axis. Suppose further that pairwise disjoint opens carry maps (1). For homogeneous $a\in\operatorname{Obs}(U_1)$ and $b\in\operatorname{Obs}(U_2)$ , compatibility with the differential includes

\begin{aligned} Q_V\mu(a\otimes b) ={}&\mu(Q_{U_1}a\otimes b)\\ &+(-1)^{|a|}\mu(a\otimes Q_{U_2}b). \end{aligned} \tag{5}

Together with the permutation and composition laws, these operations license “prefactorization.” Only the multiplicativity and Weiss comparison specified above license “factorization” in this convention. The assignment is higher-categorical only if its presentation actually supplies mapping spaces, higher transformations, or other coherent higher cells. Neither the binary operation nor one chain homotopy is such a supply by itself.

To compare this proposal with another assignment $\operatorname{Obs}'$ , a family of cochain maps

\eta_U: \operatorname{Obs}(U) \longrightarrow \operatorname{Obs}'(U) \tag{6}

must at least be natural:

j'_*\eta_U=\eta_Vj_*. \tag{7}

If disjoint operations enter the claim, it must also satisfy

\begin{aligned} \eta_V\mu^{\operatorname{Obs}}_{U_1,\ldots,U_n;V} ={}& \mu^{\operatorname{Obs}'}_{U_1,\ldots,U_n;V}\\ &\circ (\eta_{U_1}\otimes\cdots\otimes\eta_{U_n}). \end{aligned} \tag{8}

If (7) or (8) holds only up to homotopy, the chosen enhancement must supply the required coherent higher homotopies. Objectwise quasi-isomorphisms with no naturality do not form a morphism of assignments; natural objectwise quasi-isomorphisms that ignore (8) do not establish equivalence of prefactorization algebras. The model-categorical distinction between structured objectwise quasi-isomorphisms and an underived local-to-global extension is exhibited in Benini, Schenkel, and Woike 2019, Theorem 3.10, Proposition 3.13, Example 3.14, and Appendix Theorem A.1 and Example A.3.

The same type discipline applies to physical data. Suppose a chosen state and an identified scalar-field observable in each formulation produce the same two-point distribution $W_2\in\mathcal D'(M^2)$ . That is one common datum. It does not by itself prove equivalence among a Wightman theory, a Weyl net, a locally covariant functor, and a factorization algebra. States, representations, topology, causal properties, essential image, and all structure maps relevant to the conclusion still have to be compared.

Labels that remain separate in QFT

A fixed-spacetime algebraic net and a locally covariant QFT are also distinct from the factorization row. In the locally covariant formulation, a QFT is a covariant functor from a specified category of globally hyperbolic spacetimes and admissible embeddings to a category of algebras. Einstein causality and the time-slice axiom are additional conditions. Geometric covariance, causal commutation, time-slice or Cauchy constancy, additivity, and Weiss descent are different statements. The functorial formulation and the independence of Einstein causality and the time-slice axiom are explicit in Brunetti, Fredenhagen, and Verch 2003, Definition 2.1, pp. 7–8, and Proposition 2.3, pp. 10–12. In a free-field comparison, the prefactorization, Weiss, multiplicativity, and time-slice distinctions are spelled out in Gwilliam and Rejzner 2020, Definitions 2.26, 2.27, and 2.29; Remarks 2.30–2.31; and Theorems 3.5–3.6.

There are comparison theorems, but their hypotheses are part of their content. For a cocomplete closed symmetric monoidal $1$ -category target, an ordinary result identifies Cauchy-constant additive algebraic quantum field theories (AQFTs) with Cauchy-constant additive time-orderable prefactorization algebras. At the homotopical level, a 2026 paper studies theories valued in $\mathbf{Ch}_R$ , the category of cochain complexes of $R$ -modules, where $R$ is a commutative unital algebra over a characteristic-zero field. The paper restricts the spacetime category to $\mathbf{Loc}^{\mathrm{rc}}$ . For $\mathbf{Ch}_R$ -valued AQFTs and time-orderable prefactorization algebras that both satisfy the homotopy time-slice condition, it reduces the global comparison to spacetime-wise problems and proves only a spacetime-wise strictification of the AQFT time-slice condition. As of 1 August 2026, the final comparison—formulated as a spacetime-wise right Quillen equivalence—remains Open Problem 5.6. Consequently, no unrestricted equivalence between AQFTs and time-orderable prefactorization algebras may be inferred from a shared observable complex. The ordinary comparison is Benini, Perin, and Schenkel 2020, Theorem 5.1. The homotopical reduction, strictification result, and unresolved final comparison are Benini, Carmona, Grant-Stuart, and Schenkel 2026, Theorems 4.20 and 5.1, Remark 5.4, and Open Problem 5.6.

The designated specialist continuation is QFT Frameworks, Object Classes, and Typed Maps. That is where a proposed physical comparison should be classified after the present structural card has been completed.

Non-implications to keep visible

False shortcut	Witness or missing datum
complex-valued $\Rightarrow$ derived	A differential supplies no chosen weak equivalences or localization
quasi-isomorphism $\Rightarrow$ chain-homotopy equivalence	The finite $\mathbb Z/2$ example above
derived $\Rightarrow$ higher mapping data	The ordinary localization $D(R)$ does not recover a dg enhancement
higher-categorical $\Rightarrow$ derived	$\mathbf{Cat}$ has nontrivial $2$ -morphisms but no weak-equivalence localization in its bare $2$ -categorical description
technical difficulty $\Rightarrow$ higher-categorical	Higher structure requires explicit higher cells and coherence
cosheaf $\Rightarrow$ prefactorization	The compactly supported probes on the prerequisite page have no supplied disjoint multiplication
prefactorization $\Rightarrow$ factorization	The bounded-interval Weiss-cover counterexample above
ordinary Čech descent $\Rightarrow$ Weiss descent	Weiss covers use a different finite-subset condition, and homotopical descent uses the full Čech diagram
monoidal $\Rightarrow$ locality or Hilbert-space tensor factorization	A categorical tensor product supplies neither physical theorem
common observable data $\Rightarrow$ equivalent QFT frameworks	Typed maps, naturality, structured weak equivalences, and physical hypotheses remain missing

These failures are independent. Adding one missing structure does not silently add the others.

A seven-step triage procedure

When a paper or construction uses several framework labels, proceed in this order.

Fix the indexing category or site and the direction of every structural map.
List the objects, the allowed maps, and every structure those maps must preserve.
Name the weak-equivalence class $W$ and the localization, model, dg, or higher-categorical presentation in which it is used.
Identify actual higher cells and coherence; do not infer them from notation such as “up to homotopy.”
Record tensor and disjoint multi-input operations separately from any physical factorization statement.
State the precise class of covers and whether its descent comparison is strict, derived, or homotopy coherent.
Apply only the labels warranted by these entries. If the conclusion needs an existence, strictification, descent, or equivalence theorem, stop and use the specialist treatment.

For a claimed comparison $\eta:\mathcal F\to\mathcal G$ , also check the component types, differential compatibility, naturality, membership in the declared weak-equivalence class, compatibility with all multi-input operations and descent diagrams, and preservation of every physical datum used in the conclusion. If only the first four survive, say “natural objectwise quasi-isomorphism,” not “equivalent QFTs.”

Check your understanding

Label check. A construction assigns complexes and cochain inclusion maps satisfying identity and composition, with no chosen weak equivalences. Which labels are justified?

Answer check. It is a dg precosheaf. It is not yet derived, prefactorization, factorization, or higher-categorical data.

Map check. Each $\eta_U$ in (6) is a quasi-isomorphism, but (7) fails. What survives?

Answer check. There are objectwise quasi-isomorphisms, but no natural transformation of assignments and hence no structured weak equivalence.

Factorization check. The maps (1) satisfy their permutation, inclusion, and iterated-composition laws, but no cover comparison is supplied. What is licensed?

Answer check. Prefactorization only.

Higher check. A roof in $D(R)$ is displayed. Does that provide a mapping space?

Answer check. No. It represents a morphism in an ordinary derived category; a higher enhancement requires additional mapping and coherence data.

Physical check. Two formulations reproduce the same $W_2$ . What has been proved?

Answer check. Agreement on one distribution. Framework or physical equivalence remains unproved.

What this boundary map establishes

Derived, higher-categorical, and factorization structures answer different questions: which maps become equivalences, which morphism levels remain as data, and how disjoint local observables assemble. The framework card and typed comparison checks identify their intersections without collapsing them. The controlled observable assignment illustrates the strongest label licensed at each stage, while the degenerate prefactorization example proves that disjoint operations alone do not imply Weiss descent.

The page stops before constructing derived localizations or stacks, choosing models of higher categories, proving coherence or strictification, constructing factorization algebras, proving Weiss descent, developing $E_n$ -algebras or factorization homology, and establishing equivalence among QFT frameworks. Those are specialist theorems whose hypotheses must remain attached to their conclusions.

References

Marco Benini, Victor Carmona, Alastair Grant-Stuart, and Alexander Schenkel, “On the Equivalence of AQFTs and Prefactorization Algebras,” Letters in Mathematical Physics 116 (2026), article 13, DOI:10.1007/s11005-025-02035-7, Theorems 4.20 and 5.1, Remark 5.4, and Open Problem 5.6. Reduction of the homotopical comparison, spacetime-wise AQFT time-slice strictification, and the unresolved spacetime-wise right-Quillen-equivalence problem.
Marco Benini, Marco Perin, and Alexander Schenkel, “Model-Independent Comparison Between Factorization Algebras and Algebraic Quantum Field Theory on Lorentzian Manifolds,” Communications in Mathematical Physics 377 (2020), 971–997, DOI:10.1007/s00220-019-03561-x, Theorem 5.1. The ordinary categorical comparison under additivity, time-orderability, and time-slice hypotheses.
Marco Benini, Alexander Schenkel, and Lukas Woike, “Homotopy Theory of Algebraic Quantum Field Theories,” Letters in Mathematical Physics 109 (2019), 1487–1532, DOI:10.1007/s11005-018-01151-x, Theorem 3.10 for a ground field containing $\mathbb Q$ , Proposition 3.13, Example 3.14, and Appendix Theorem A.1 and Example A.3. Structured objectwise quasi-isomorphisms, model structures, and the failure of underived local-to-global extension to preserve weak equivalences.
Romeo Brunetti, Klaus Fredenhagen, and Rainer Verch, “The Generally Covariant Locality Principle: A New Paradigm for Local Quantum Field Theory,” Communications in Mathematical Physics 237 (2003), 31–68, arXiv:math-ph/0112041, Definition 2.1, pp. 7–8, and Proposition 2.3, pp. 10–12. Covariant functoriality, Einstein causality, and the time-slice axiom as separate conditions, and the passage from a locally covariant theory to a fixed-spacetime net.
Kevin Costello and Owen Gwilliam, “Factorization Algebra” (2023), arXiv:2310.06137v2, Definitions 1–3 and Remarks 1–2, pp. 2–3 and 10–12. Disjoint multi-input operations, Weiss covers, multiplicativity, and strict versus derived descent.
Owen Gwilliam and Kasia Rejzner, “Relating Nets and Factorization Algebras of Observables: Free Field Theories,” Communications in Mathematical Physics 373 (2020), 107–174, DOI:10.1007/s00220-019-03652-9, Definitions 2.26, 2.27, and 2.29; Remarks 2.30–2.31; and Theorems 3.5–3.6. A hypothesis-sensitive comparison of free-field constructions; prefactorization, Weiss, multiplicativity, and time-slice distinctions. Remark 2.30 records an analytic colimit condition not proved there.
Jacob Lurie, Higher Algebra, version dated September 18, 2017, author’s PDF, Example 1.1.1.12, §§ 1.3.2 and 1.3.5, and § 5.4.5. Stable and derived $\infty$ -categories, localization at quasi-isomorphisms, and the scoped locally constant factorization/ $E_n$ relationship.
Emily Riehl and Dominic Verity, Elements of ∞-Category Theory, Cambridge University Press (2022), preface pp. ix–xi and Definitions 1.1.2 and 1.1.10, DOI:10.1017/9781108936880. Higher morphisms as structure, homotopy coherence, models of $\infty$ -categories, and equivalence.
The Stacks Project Authors, Derived Categories, The Stacks Project (continuously updated; accessed August 11, 2026), Stacks Project Authors, § 13.11, tag 05RR, especially Definition 13.11.3. Localization of the homotopy category at quasi-isomorphisms and the classical derived category.