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On dualizability and invertibility in the higher Morita category

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read An E_n-algebra in the Morita category is (n+1)-dualizable exactly when it is left dualizable as a module over each sphere-shaped factorization homology.

desk verdict A serious, technically advanced preprint that plausibly proves Lurie's (n+1)-dualizability conjecture and the BJSS invertibility corollary, but the central proof is conditional on unpublished results identifying the new pointless factorization model with the Morita category. read the letter →

arxiv 2607.16953 v1 pith:RCKRNLNZ submitted 2026-07-18 math.CT math.ATmath.QA

Claudia Scheimbauer , Pelle Steffens , William Stewart This is my paper · ORCID
classification math.CTmath.ATmath.QA MSC 18N7057K16
keywords higherMoritacategoriesdualizabilityE_n-algebrasfactorizationhomologyalgebrastopologicalfieldtheoriesadjointabilityrelative
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a long-standing conjecture characterizing when an algebra with n compatible multiplications is (n+1)-dualizable in the higher Morita category: this happens precisely when the algebra is dualizable as a left module over each of n+1 explicitly defined sphere-shaped factorization homologies. Because of the Cobordism Hypothesis, such dualizable objects are exactly the inputs for fully extended framed topological field theories, so the result converts a finite set of module-duality conditions into a large family of field theories. The proof works by rebuilding the higher Morita category from 'pointless' factorization algebras, which remove unwanted basepoint data that had blocked earlier approaches, and then using a critical-point geometric argument to identify the bimodules witnessing dualizability with the required factorization homologies. A corollary gives a criterion for invertibility in the Morita category, and a relative version produces lax and oplax relative field theories.

What carries the argument

The load-bearing new object is the pointless higher Morita category, built from constructible 'pointless' factorization algebras on stratified cubes: objects are E_n-algebras, k-morphisms for k < n are iterated bimodules, and n-morphisms are bimodules without chosen elements. The absence of pointings is what makes (n+1)-dualizability possible, since previously the pointings forced all sufficiently dualizable objects to be trivial. Two supporting mechanisms carry the proof: a lifting-of-adjoints theorem that reduces adjointability of an n-morphism to adjointability of an underlying ordinary bimodule, and a critical-point geometric computation identifying the bimodules that witness n-dualizabi

What would settle it

Take an E_n-algebra A whose factorization-homology modules over S^{k-1} × R^{n-k+1} are all left dualizable and compute its (n+1)-dualizability in an independent, established model of the higher Morita category; if the dual fails there, the criterion is false as a statement about the standard Morita category. A low-dimensional check is the n=1 or n=2 case, where the criterion should reproduce the known classifications of 2-dualizable algebras and 3-dualizable tensor categories; a mismatch, or a failure of the model's composition to be the relative tensor product, would falsify the paper's cent

Watch

Extended reading notes

Core claim

The central claim is Theorem A: in the Morita (∞, n+1)-category of a presentably symmetric monoidal ∞-category, an E_n-algebra A is (n+1)-dualizable if and only if for every 0 ≤ k ≤ n, A is left dualizable as a left module over the factorization homology of A over S^{k-1} × R^{n-k+1}. This is the missing characterization of high dualizability: the right-to-left direction says that finitely many module-dualizability conditions over explicit geometric spaces suffice to produce all the higher adjoints that dualizability requires. In particular, by the Cobordism Hypothesis, such an A gives a fully extended framed (n+1)-dimensional topological field theory that assigns A to the framed point.

Load-bearing premise

The load-bearing premise is that the newly built pointless category is genuinely the higher Morita category of E_n-algebras—that its k-morphisms are iterated bimodules and its composition is the relative tensor product; the proofs of these identifications are deferred to later work, and the equivalence with the established combinatorial model is not proved here.

Editorial extensions

If this is right

  • Every (n+1)-dualizable E_n-algebra in a nice symmetric monoidal ∞-category yields, via the Cobordism Hypothesis, a fully extended framed (n+1)-dimensional topological field theory valued in the Morita category.
  • An E_n-algebra is invertible in the Morita category exactly when it is (n+1)-dualizable and each sphere factorization homology maps canonically to the corresponding E_{n-k}-center as an equivalence.
  • The relative theorem gives a criterion for a morphism between E_n-algebras to be n-times right adjointable, in terms of dualizability over certain stratified half-sphere factorization homologies, producing oplax relative fully extended field theories.
  • The two standard kinds of relative boundary theories arise uniformly: the regular module from the unit always gives an oplax relative theory, while the reverse module gives one exactly when it is n-dualizable in the lower Morita category.
  • The dualizability and invertibility results extend to Morita categories valued in higher (∞, d)-categories, because the relevant properties are detected in the underlying (∞, n+1)-category after truncation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper defers the proof that composition in the pointless model is the relative tensor product and leaves the comparison with the established combinatorial model open; if that identification fails, Theorem A is a dualizability statement about a new category rather than the Morita category in the original conjecture.
  • The criterion is finite and explicit, so it could be used as a test for dualizability in concrete algebraic examples (e.g., braided tensor categories and fusion 2-categories), potentially revealing which familiar objects admit fully extended framed TFTs.
  • The success of 'unpointing' suggests a general principle: basepoint data in factorization models of higher categories obstruct dualizability, and marking-preserving inclusions may be a broadly applicable remedy in other higher-categorical settings.
  • The relative criterion likely produces more twisted field theories than the two boundary cases named here; any morphism whose half-sphere factorization-homology modules are dualizable should give an interface between the absolute theories of its source and target.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a new model uMor_n(C) for the higher Morita category of E_n-algebras, based on 'pointless' constructible factorization algebras on marked stratified cubes. Its main result, Theorem 3.0.1, asserts that an E_n-algebra A in this category is (n+1)-dualizable if and only if, for every 0 ≤ k ≤ n, A is left dualizable as a left module over the factorization homology ∫_{S^{k-1} × R^{n-k+1}} A. From this it derives the invertibility criterion of Corollary 3.5.1 and a relative dualizability theorem, Theorem 4.1.3, aimed at relative/twisted field theories. The proof combines a lifting-of-adjoints argument (Section 3.1) with a Morse-theoretic computation (Section 3.3), building on previous work of the first author and collaborators.

Significance. If the central proof is completed, the paper would prove a long-standing conjecture of Lurie (Remark 4.1.27 of [Lur09b]) and the Brochier–Jordan–Safranov–Snyder invertibility conjecture, and would supply a broad framework for fully extended framed TFTs with Morita-type targets. The paper also contains several auxiliary results that are valuable in their own right: the operadic localization theorem 2.3.15, the Segal condition for the pointless factorization model (Proposition 2.4.40), and the identification of pointless constructible factorization algebras on (R,0) with bimodules (Corollary 2.3.18). The n=1 case of the main criterion matches Lurie's known bimodule adjoint criterion [Lur17, Prop. 4.6.2.13]. However, the significance is conditional: the category uMor_n is not yet shown to be the accepted higher Morita category of E_n-algebras, and several load-bearing structural results are deferred to unpublished work.

major comments (2)
  1. [§2.4.5, Propositions 2.4.42 and 2.4.43] The central construction and proof depend on two results that are explicitly deferred to [SSS]: Pointless additivity (Prop. 2.4.42) and the identification of composition with the relative tensor product (Prop. 2.4.43). These are not optional refinements: they are used to prove quasi-unitality (Prop. 2.4.44), the Cartesian fibration structure of morphism categories (proof of Prop. 2.4.61), the properties of regular bimodules (Prop. 2.4.64 and Cor. 2.4.69), and ultimately the lifting-of-adjoints theorem 3.1.3 and Theorem 3.0.1. Without these propositions, Theorem A is a dualizability statement for a newly assembled category whose k-morphisms have not been identified with iterated bimodules and whose composition has not been identified with Morita composition. This is a load-bearing gap. The same concern applies to the relative Theorem B. The authors should either include full proofs of Pro
  2. [Remark 3.1.1] The paper concedes that a precise comparison between uMor_n(C) and Haugseng's combinatorial higher Morita category is beyond its scope. Since the conjecture of Lurie that the paper claims to prove is about the Morita category of E_n-algebras, a dualizability theorem for a model that is not compared to an established model does not by itself settle the conjecture. If the intended contribution is to prove the conjecture, the equivalence with Haugseng's model (or with another accepted model) must be established in this paper or replaced by a precise reference with a complete proof. Otherwise the claim 'we prove a conjecture of Lurie' in the abstract is premature.
minor comments (4)
  1. [Title] The title contains apparent spelling errors: 'INVER TIBILLITY' and 'CA TEGOR Y' should be corrected.
  2. [Throughout] The text contains many corrupted or overlapping Unicode arrow symbols such as '/leftr⫯g⊸tl⫯ne→' and '⫯ne→'. These should be cleaned up before publication.
  3. [Introduction, Theorem B] The introduction states 'Theorem 4.1.3 and Corollary 4.2.7' and adds 'even = oplax case'. The relationship between the main relative theorem and the lax/oplax terminology could be clarified, especially whether the lax case is also proven.
  4. [Theorem 3.0.1] Since the theorem is stated for both Mor_n(C) and uMor_n(C), it may help to add a short reminder that univalent completion does not affect dualizability; Remark 2.4.52 already explains this, but a sentence in the main theorem would reduce potential confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

Dualizability proof is independent, but the paper's identification of uMorn with the Morita category is deferred to the authors' own [SSS] and a comparison with Haugseng's model is explicitly not proved.

  1. self citation load bearing [§2.4.5, Propositions 2.4.42–2.4.43; §3.1, Remark 3.1.1]
    "Proposition 2.4.42 (Pointless additivity) ... A proof will appear in upcoming joint work of the first two authors with Anja Švraka [SSS]. ... Proposition 2.4.43 ... A proof will appear in upcoming joint work of the first two authors with Anja Švraka [SSS]. ... Remark 3.1.1: While it is the case that the (pointless) factorization Morita category of Section 2.4 and the combinatorial Morita category of [Hau17] are equivalent, a precise comparison is beyond the scope of this article..."

    Theorem A's status as a proof of Lurie's conjecture is about the higher Morita category of E_n-algebras. That interpretation depends on Proposition 2.4.42 (k-morphisms are iterated bimodules) and Proposition 2.4.43 (composition is relative tensor product), both deferred to [SSS], a paper by the same authors. Remark 3.1.1 concedes that the equivalence with Haugseng's combinatorial Morita category is not proved. Thus the central premise linking uMorn(C) to the conjectured Morita category is not derived here but imported from the authors' unpublished future work; if that work fails, Theorem A is a dualizability theorem about a new model rather than about Lurie's Morita category. The dualizability computation itself is not circular, but the claim to resolve the conjecture is load-bearing on se

full rationale

The main derivation of (n+1)-dualizability is not circular: the criterion is obtained from GS18's n-dualizability, the lifting-of-adjoints theorem, a Morse-theoretic identification of the relevant bimodules, and Lurie's 1-dimensional bimodule adjoint criterion. None of these inputs is the theorem's conclusion, and the final condition (left dualizability over the relevant factorization homologies) is a genuine, non-vacuous condition rather than a renamed assumption. The invertibility corollary likewise follows formally from the main theorem plus a center-detection statement, not from assuming the Brochier–Jordan–Safranov–Snyder conjecture. However, the paper's advertised interpretation as proving Lurie's conjecture depends on two deferred propositions and a missing comparison with Haugseng's model; both are tied to the authors' own upcoming [SSS]. This is a load-bearing but non-circular infrastructure gap, so the score is 4 rather than 0; there is no equation-level reduction of the theorem to its inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

No free parameters: the theorem is an unconditional criterion for all E_n-algebras in any presentably symmetric monoidal ∞-category; nothing is fitted or tuned. The axiom burden is concentrated in infrastructure: the pointless factorization algebra framework ([KS], [KSW26], by the same research group), n-dualizability from [GS18] (first author's prior work), and two propositions (2.4.42, 2.4.43) deferred to the unpublished [SSS]. The target conjectures of Lurie and Brochier–Jordan–Safranov–Snyder are external and are not assumed, so the ledger records heavy self-reliance rather than circularity. The genuine invented entities are models, not speculative physical objects; their independent status is pending the Haugseng-model comparison.

assumptions (8)
  • ad hoc to paper Pointless additivity (Prop 2.4.42): pushforward along the projection ◻^n_[k]→◻_[k] identifies Fact^cstr on the product with Fact^cstr on the line valued in Fact^cstr on the remaining factor
    Stated in §2.4.5 with 'A proof will appear in upcoming joint work of the first two authors with Anja Švraka [SSS]'. Needed to identify k-morphisms of uMor_n with iterated bimodules (the claim that the model is the higher Morita category) and for quasi-unitality (Prop 2.4.44). Unproven in this preprint.
  • ad hoc to paper Composition in uMor_n is the relative tensor product of bimodules (Prop 2.4.43)
    Deferred to [SSS] ('A proof will appear...'). The paper sketches the underlying-object computation via a Weiss cover, but the functor-level statement used in Prop 2.4.64(5) and the adjoint arguments is not proven here. Needed for quasi-unitality and for the geometric argument behind Theorem A.
  • domain assumption Gluing/descent (sheaf property) for pointless constructible factorization algebras on marked smooth conical manifolds
    Used for the Segal condition (Prop 2.4.40), citing [KS, Theorem 5.3] and [KSW26]; foundational input from closely related work of the same authors (Karlsson–Scheimbauer, Karlsson–Scheimbauer–Walde).
  • domain assumption Every E_n-algebra is n-dualizable and all k-morphisms (k<n) are adjointable in the pointed Morita category ([GS18])
    Invoked in §1.4.2: 'In [GS18] it was shown that every E_n-algebra A is n-dualizable as an object in the pointed Morita category'. Supplies the (co)evaluation and unit/counit data on which the (n+1)-dualizability step of Theorem A is built; prior work of the first author.
  • standard math A bimodule _A M_B of E_1-algebras admits a left adjoint iff M is left dualizable over A ([Lur17, Prop 4.6.2.13])
    Explicitly used (§1.4.2) to convert the geometric description of the n-morphism bimodules (Prop 3.3.7) into the factorization-homology dualizability criterion of Theorem A.
  • domain assumption Cobordism Hypothesis (Baez–Dolan–Lurie): k-dualizable objects in a symmetric monoidal (∞,k)-category classify framed fully extended k-dimensional TFTs
    Used in §1.1 to pass from Theorem A to existence of fully extended framed (n+1)-dimensional TFTs and in §4 (via [JFS17]) for relative field theories.
  • domain assumption Stratified cubes ◻^n_[k] admit factorizing disk-bases ('enough good disks')
    The paper verifies this for its cubes (Cor 2.4.33). Needed for equivalence between constructible pointless factorization algebras and disk-algebras (Cor 2.3.16), which underpins the whole model. Remark 2.2.25 notes it is open whether every marked smooth conical manifold has such a basis.
  • ad hoc to paper The pointless model uMor_n(C) is equivalent to Haugseng's combinatorial higher Morita category
    Explicitly not proven: Remark 3.1.1 says 'a precise comparison is beyond the scope of this article'. Without it, Theorem A is a statement about the new model, and transferring it to the established Morita category (the one Lurie's conjecture concerned) is an unverified assumption.
invented entities (2)
  • Pointless (constructible) factorization algebras on marked smooth conical manifolds (X,E), E a subset of the 0-dimensional strata
    purpose: Remove the 'pointing' (chosen element) that made objects of the pointed factorization model only trivially (n+1)-dualizable (see [JF19], [GS18] §1.2), enabling the proof of (n+1)-dualizability in the Morita category.
    Framework introduced in [KS] (Karlsson and first author), extended here with the localization theorem (2.3.15). Internal checks: Cor 2.3.18 shows (R,0)-pointless FAs are exactly bimodules, and the n=1 reduction of Theorem B recovers Lurie's known criterion. External confrontation is incomplete: the equivalence with Haugseng's model is deferred to [SSS] (Remark 3.1.1).
  • The higher Morita category uMor_n(C) of pointless factorization algebras (Definition 2.4.51)
    purpose: A purpose-built model of the Morita (∞,n+1)-category of E_n-algebras in which geometric manipulations (pushforwards along collapse-rescale and Morse maps) are available for dualizability arguments.
    Internal consistency is established in the paper (Segal condition 2.4.40, quasi-unitality 2.4.44 up to deferred additivity, bimodule identification 2.3.18). Independent status depends on the deferred comparison to Haugseng's model and the deferred propositions 2.4.42–2.4.43.

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Pith. "Pith review of On dualizability and invertibility in the higher Morita category." pith.science (2026). https://pith.science/paper/RCKRNLNZ

@misc{pith2026260716953,
  author       = {Pith},
  title        = {Pith review of: On dualizability and invertibility in the higher Morita category},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCKRNLNZ}},
  note         = {Machine review of arXiv:2607.16953}
}
abstract

We prove a conjecture of Lurie characterizing $(n+1)$-dualizability in higher Morita categories of $\mathbb{E}_n$-algebras in terms of dualizability over certain factorization homologies. A key ingredient is a higher Morita category based on the recently developed framework of pointless factorization algebras of Karlsson and the first author. We also verify an invertibility conjecture of Brochier--Jordan--Safranov--Snyder as an immediate corollary of our main result. Moreover, we prove a relative version of the dualizability conjecture, yielding a new criterion for relative/twisted field theories. We give some examples, including Dirichlet and Neumann relative theories.

Figures

Figures reproduced from arXiv: 2607.16953 by the authors.

Figure 1
Figure 1. The four Morse foliations in a neighborhood of (1/2, 1/2) correspond￾ing, from left to right, to the functions g 0,0 , g 1,0 , g0,1 and g 1,1 . whose derivative is (dgw ′ (x) sin(πy),−π(1 − g w ′ (x)) cos(πy)), which vanishes only when y = 1/2 and x = (1/2, ..., 1/2). At this point the Hessian is given by [ Hgw′ (1/2, ..., 1/2) 0 0 π 2 2 ] which has index ∣w ′ ∣ = ∣w∣. Similarly, if wk = 1, then in a neighborhood of… view at source ↗

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.