{"id":"388b5054-9dd6-4e2d-88f5-5307a0d55d59","arxiv_id":"2411.19904","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Integrals with variable upper limits are re-encoded as integral posets, and elementary functions and gentle-algebra global dimensions are rewritten in this framework, largely restating known results.","lead":"The paper recasts integrals with variable upper limits as partially ordered sets of pairs (interval, integral value) built from normed modules over finite-dimensional algebras. A generalist might read it to see whether categorical integration can connect classical analysis with representation theory, but most results repackage earlier work by the same authors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.8 assumes bT(1_S)=µ(S) for all measurable S, a property not stated in Theorem 2.17; this makes the surjectivity of ⅁♮∘⅁ and hence Theorem 3.14 unsupported for general A^p.","rationale":"The reader's weakest assumption identifies the reliance on the authors' prior results [21], especially Theorem 2.16 (initial object) and Theorem 2.17 (integral realization), and notes that Lemma 3.8's surjectivity construction would collapse if those theorems fail. My analysis sharpens this: even assuming Theorem 2.17(1) exactly as quoted, it does not supply the identity bT(1_S)=µ(S) for arbitrary measurable S, which Lemma 3.8 silently uses. This is a distinct, more specific unproven step that directly threatens Theorem 3.14 and therefore the main categorification theorems that depend on it. However, the concern is repairable: under the L-conditions, bT is the Lebesgue integral and bT(1_S)=µ(S) is standard, so Corollary 4.3 and the concrete variable-upper-limit interpretation may survive. The paper could fix the gap by either proving the missing property for general A^p or explicitly restricting Lemma 3.8/Theorem 3.14 to the L-condition setting. Because the verdict CONDITIONAL already anticipates such repairs, my read does not change the verdict, but it makes the required repair more specific.","tokens_in":33814,"tokens_out":9846,"duration_ms":85465,"concrete_test":"Check the proof of Theorem 7.6 in the authors' previous work [21] to see whether it establishes bT_{k,µ(IΛ),m}(1_S) = µ(S) for every S ∈ Σ(IΛ), not just S = IΛ. Independently, in a concrete instance (e.g., Λ = k, τ = id, p = 1, I = [0,1], S = [0,1/2]), compute the unique morphism from the initial object to (k, µ(IΛ), m) directly from the initial-object definition and compare bT(1_S) with µ(S). If bT(1_S) ≠ µ(S) for some such S, Lemma 3.8's construction collapses and Theorem 3.14 needs either an additional hypothesis or a restriction to the L-condition setting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central categorical claim rests on Theorem 3.14, whose proof depends on Lemma 3.8. In Lemma 3.8, for an arbitrary measurable S ⊆ IΛ, the authors define f = k/µ(S) · 1_S and assert bT_{k,µ(IΛ),m}(f) = k/µ(S) · µ(S) = k, i.e., they use bT_{k,µ(IΛ),m}(1_S) = µ(S). However, Theorem 2.17(1), the only stated characterization of bT_{k,µ(IΛ),m}, guarantees this identity only for S = IΛ: it says bT(1_IΛ) = µ(IΛ), k-linearity, and an inequality, but nothing about arbitrary measurable subsets. The equality bT(1_S) = µ(S) is a stronger property that is true for the Lebesgue integral under the L-conditions, but Lemma 3.8 is stated before the L-conditions are imposed and Theorem 3.14 is claimed for general A^p. Without this missing property, the proof that ⅁♮⅁ is surjective fails: there is no reason that the abstract morphism from the initial object to (k, µ(IΛ), m) assigns measure to the indicator of every measurable set. Consequently, the Λ-epimorphism statement of Theorem 1.1/3.14 is not established as written for the general framework; it is only clearly valid under the L-conditions where bT is known to be Lebesgue integration. This gap is load-bearing because Theorem 3.17 and Corollary 4.3 rely on Theorem 3.14 for linearity and the module structure, even if the concrete variable-upper-limit formula may survive under L-conditions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a categorical framework for integrals with variable upper limits in the spirit of Leinster's A^p categories, building on the authors' prior work on normed modules over finite-dimensional algebras. It introduces integral partially ordered sets bTI(f), an addition rule (3.3) on pairs of the form ([u,v], bT_v^u(f)), and maps ⅁, ⅁♮ relating these pairs to elements of Σ(IΛ)×k. The central results are Theorem 3.14 (the restriction of ⅁♮ to Im(+) is a Λ-epimorphism), Theorem 3.17 (additivity and k-linearity of integrals with variable upper limits under L-conditions), and Corollary 4.3, which purports to recover the Leinster-Meckes categorification of the Lebesgue integral with variable upper limit. Two applications are offered: a categorified account of elementary functions (trigonometric, logarithmic, exponential) and new formulas for the global dimension of gentle algebras in terms of multiple and Lebesgue-Stieltjes integrals.","tokens_in":34149,"tokens_out":9926,"duration_ms":80718,"significance":"If the main theorems were fully established, the paper would contribute a novel categorical language for the map t ↦ ∫_α^t f dµ and its additivity, connecting Leinster's categorification to variable upper limits. The authors are systematic and careful in verifying the k-linear and Λ-module structures on the images of + and ⅁ (Lemmas 3.5, 3.11), and they clearly state their dependence on the companion papers [20,21]. However, as documented below, several load-bearing points are not proved as written, and one application uses an invalid base field. The genuinely new content beyond the quoted results of [20,21] and the definitions is currently limited, and the advertised scope of the main theorems needs substantial revision.","major_comments":[{"comment":"The proof of Lemma 3.8 defines f = k/µ(S) · 1_S and asserts bT_{k,µ(IΛ),m}(f) = k, which requires the identity bT(1_S) = µ(S) for arbitrary measurable S. Theorem 2.17(1) only states bT(1_{IΛ}) = µ(IΛ), k-linearity, and the inequality |bT(|f|)| ≤ |bT(f)|; it does not determine bT on arbitrary measurable subsets. This equality is known for the Lebesgue integral under the L-conditions, but Lemma 3.8 is stated before L-conditions are imposed and Theorem 3.14 is claimed for all A^p. Moreover, the construction is undefined when µ(S)=0. Consequently, the surjectivity of ⅁♮⅁, and hence the Λ-epimorphism assertion of Theorem 3.14, is not established for the general framework.","section":"§3.3, Lemma 3.8 and Theorem 3.14"},{"comment":"The proof begins: 'by Proposition 3.13, we have known that Im(+) is a left Λ-submodule of Im(⅁)'. This reverses the actual statement of Proposition 3.13, which says Im(⅁) is a left Λ-submodule of Im(+). The proof then verifies Λ-linearity but does not prove surjectivity; the final sentence 'Since dim_k({IΛ}×k)=1, we find that ⅁♮|Im(+) is a Λ-epimorphism' is a non sequitur unless one already knows the map is nonzero or surjective, which would require an appeal to Lemma 3.8 (whose proof is incomplete, as noted above).","section":"§3.3, proof of Theorem 3.14"},{"comment":"The additivity identity in Theorem 3.17(1) is a direct consequence of the definition of the addition + in (3.3). The proof invokes Lemma 3.4, and that lemma is proved immediately by unpacking the cases of (3.3). Thus the identity holds by construction rather than being a substantive consequence of the categorical structure. The paper should either present this as a consistency check of the definition of + or provide an independent argument showing why the definition is forced by the categorical framework; the current framing overstates the content.","section":"§3.4, Theorem 3.17(1)"},{"comment":"The base object k = Λ = R×[0,2π) with multiplication (r1,θ1)·(r2,θ2)=(r1r2,θ1+θ2 mod 2π) is not a normed field containing R. For any θ>0, (0,θ)·(1,2π−θ)=(0,0), so nonzero zero divisors exist, and no field addition is specified on R×[0,2π). Since Theorem 2.17 and the definition of A^p require k to be a complete field, the derivation of K=π/2 and the identification of the functions s and c with sine and cosine are not justified.","section":"§5.1.2, conditions (2')–(6')"},{"comment":"The global-dimension formulas are, in substance, restatements of the authors' earlier result [20, Theorem 5.10] (quoted in (5.8)) together with the elementary computation (5.7), which essentially rewrites the length ℓ(q) as 2∫v_q dµ. The Stieltjes-integral formula in Theorem 5.6 is not rigorously derived: φ_l in (5.13) is written as a function x↦ln x^l rather than as a measure, and the chain of equalities leading to (L-S)∫_1^2 w_P|kP dφ_l = l omits the measure-substitution hypotheses. The claims of a 'new approach' for characterizing global dimensions should be toned down, and the derivations should be completed.","section":"§5.4–5.5, Theorems 5.5 and 5.6"}],"minor_comments":[{"comment":"The symbol ⅁ is used for several different maps (⅁, ⅁♮, ⅁α, ⅁α,f) with closely related meanings; more mnemonic or indexed notation would greatly improve readability.","section":"General notation"},{"comment":"The definition of elementary simple functions writes f(k1,...,kn)=Σ k_i 1_{X_i}, but the domain IΛ is a subset of Λ; the meaning of the coordinates k_i and the sets X_i should be clarified.","section":"§2, Definition 2.7"},{"comment":"The definition of ⅁α uses the condition α ∈ (c,d)k, but the set Sα,f is then described for α ≤ t ≤ d, which includes the case α=c; please make the domain consistent.","section":"§3.3.1, equation (3.5)"},{"comment":"The statement '⅁ is an injection' is made for the map on the disjoint union S over all α and f; the proof only considers equality of images of two elements, which is fine, but the domain of ⅁ should be stated explicitly to avoid confusion with the restriction ⅁α.","section":"§3.3.1, Lemma 3.10"},{"comment":"The computation contains the expression ∫_0^t k_{e_v} d k_{e_v} = t^2/2, which is nonstandard notation; it should be rewritten as an ordinary integral over a real variable with a clear substitution.","section":"§5.3.2, computation of bTI(v_q)"},{"comment":"There are numerous typographical errors and infelicities (e.g., 'defied' for 'defined', 'structrue' for 'structure', inconsistent use of k and R). The manuscript would benefit from a careful proofreading pass.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on the authors' own preprints [20] and [21], including the initial-object theorem (Theorem 2.16) and the global-dimension formula (5.8). The editor should confirm that those companion works are available in stable form. The treatment of R×[0,2π) as a field in Section 5.1.2 appears to confuse complex multiplication with the quotient by a period; this should be raised explicitly with the authors. The central theorem 3.14 may be salvageable by restricting to the L-conditions and proving the missing property bT(1_S)=µ(S), but as written the general claim is unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: Sections 3–4 build a module-theoretic language for variable-upper-limit integrals and recover Leinster–Meckes; the framework is mostly honest, but the main additivity statement is built into definition (3.3), and the two applications are not reliable as printed. I would send this to a referee, but I would expect heavy revision.\n\nWhat is new: integral posets bTI(f), the maps ⅁⅁⅁ and ⅁⅁⅁♮, the addition rule (3.3), and formula (5.7) expressing a path length as twice an integral of a vertex homomorphism. Corollary 4.3 genuinely recovers the Leinster–Meckes result. The module structures on Im(+) and Im(⅁) check out in broad strokes, and the authors are careful to cite their prior work [20,21] for the load-bearing initial-object and global-dimension facts.\n\nSoft spots, in order of importance.\n\n1. Lemma 3.8 overreaches. Its proof assumes bT(k,µ(IΛ),m)(1_S)=µ(S) for arbitrary measurable S, but Theorem 2.17 gives that identity only for S=IΛ. It is true for Lebesgue measure under the L-conditions, but Lemma 3.8 and Theorem 3.14 are stated for general A^p and they need that surjectivity. The division by µ(S) also silently assumes nonzero measure. So Theorem 1.1/3.14 is not established as stated; this is a repairable gap, not a fatal one.\n\n2. The proof of Theorem 3.14 has a containment typo: it says Im(+) is a submodule of Im(⅁), which is the reverse of Proposition 3.13. The epimorphism conclusion also leans on dim_k=1 plus the unproved surjectivity. Minor once Lemma 3.8 is fixed under L-conditions.\n\n3. The additivity Theorem 3.17(1) is close to definitional: rule (3.3) was chosen so interval-telescoping holds, and Lemma 3.4 is immediate. That does not make it false, just less exciting than the abstract suggests.\n\n4. Application 1: R×[0,2π) with angle-pointwise multiplication is not a field, so it cannot serve as the base field k. The derivation of 2K=π uses the known circumference 2πR and the known value of ∫_{-1}^1 (1-t²)^(-1/2) dt; π is not derived, it is assumed. The displayed chain is also inconsistent with the conclusion K=π/2.\n\n5. Application 2 is a translation of the authors' own gl.dim A = sup forbidden-thread length into integral language. Formula (5.7) is nice, but the claim of a new approach is overstated.\n\nBottom line: the core framework is plausible and deserves referee time, but the paper should not be accepted until Lemma 3.8 is repaired or qualified, the trig section is rewritten or cut, and the provenance of the global-dimension characterization is acknowledged. Right now it is a conditional accept with major revision.","headline":"A defensible but largely definitional categorification of variable-upper-limit integrals, with two applications that do not hold up as printed; worth refereeing for the core Sections 3–4, not for the trig/global-dimension claims.","tokens_in":834,"tokens_out":1176,"would_cite":false,"duration_ms":44137,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","46B99","46M40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that variable-limit Lebesgue integration is categorified by a category of Banach modules, with the classical addition formula as a corollary.","keywords":["finite-dimensional algebras","normed modules","categorification","integrals with variable upper limits","integral partially ordered sets","gentle algebras","global dimension","Lebesgue-Stieltjes integrals"],"falsifier":"Compute both sides of Theorem 3.17(1) for $\\Lambda=\\mathbb{R}$, $c=0$, $d=3$, $\\alpha=0$, $\\beta=1$, $\\gamma=2$, and $f(x)=x$ using ordinary Lebesgue integrals; if the categorical sum prescribed by equation (3.3) does not equal $([0,2],3/2)$, the additivity theorem fails. Also check whether $k=\\mathbb{R}\\times[0,2\\pi)$ with multiplication $(r_1,\\theta_1)(r_2,\\theta_2)=(r_1r_2,\\theta_1+\\theta_2 \\bmod 2\\pi)$ is a field: the element $(0,\\pi/2)$ has no inverse, so the trigonometric setup of Section 5.1.2 does not satisfy the paper's normed-field axioms.","tokens_in":33471,"feed_emoji":"∫","tokens_out":14239,"duration_ms":114929,"temperature":0.7,"pith_summary":"The paper sets out to prove that integrals with variable upper limit—the map $t \\mapsto \\int_c^t f\\,d\\mu$—can be viewed as living inside the category $A^p_\\Lambda$ of integral Banach modules over a finite-dimensional algebra $\\Lambda$. Its central construction collects pairs $([t_1,t_2]_k, \\widehat{T}_{t_2}^{t_1}(f))$ into an integral partially ordered set, defines an addition on such collections, and shows that the map that forgets the interval is a $\\Lambda$-epimorphism from the image of this addition onto $\\{I_\\Lambda\\}\\times k$ (Theorem 3.14). Under explicit L-conditions the same machinery yields the classical additivity identity $\\int_\\alpha^\\beta f+\\int_\\beta^\\gamma f=\\int_\\alpha^\\gamma f$ (Theorem 3.17) and recovers the known categorical description of the Lebesgue integral (Corollary 4.3). If correct, this makes the addition of integrals with different upper limits a piece of algebra rather than an analytic fact, and it gives categorical definitions of trigonometric, logarithmic, and exponential functions plus integral formulas for the global dimension of gentle algebras.","feed_headline":"A category that adds variable-limit integrals","feed_subtitle":"Integral posets turn the classical additivity of Lebesgue integrals into one epimorphism.","key_machinery":"The load-bearing object is the integral partially ordered set $\\widehat{T}_I(f)$: for $f\\in \\widehat{S}_\\tau(I_\\Lambda)$ it is the collection of pairs $([t_1,t_2]_k,\\widehat{T}_{t_2}^{t_1}(f))$ ordered by inclusion of intervals. The addition on these sets is defined by the four cases of equation (3.3), which cut overlapping or adjacent intervals and add the integrals of the restricted functions; the image of this addition carries a left $\\Lambda$-module structure whose action is $(a,(S,r))\\mapsto (S,\\tau(a)r)$. The maps $\\widehat{\\int}$ and its interval-forgetting companion translate between the analytic data $(t,\\int_{[\\alpha,t]_\\Lambda}f\\,d\\mu)$ and the categorical data $([\\alpha,t]_k,\\widehat{T}_t^\\alpha(f))$. What makes the argument run is initiality: because $\\widehat{S}_\\tau(I_\\Lambda)$ is initial, each $\\widehat{T}_{t_2}^{t_1}(f)$ is the restriction of the integral over any larger interval, so adding two consecutive interval integrals in the category automatically produces the integral over the union, and the classical additivity law falls out as Theorem 3.17(1).","core_discovery":"On the paper's own terms, the discovery is that the addition of variable-upper-limit integrals is controlled by the initial object of $A^p_\\Lambda$. For a finite-dimensional algebra $\\Lambda$ with basis $b_i$ and an interval $I=[c,d]_k$, the completed space $\\widehat{S}_\\tau(I_\\Lambda)$ of elementary simple functions on $I_\\Lambda=\\sum_i [c,d]_k b_i$ is, by a theorem quoted from [21], initial in $A^p_\\Lambda$; every object therefore receives exactly one morphism $\\widehat{T}_{(N,v,\\delta)}$. The paper's new move is to take the family $\\{([t_1,t_2]_k,\\widehat{T}_{t_2}^{t_1}(f))\\}$ as an integral partially ordered set, to define an addition on these families by gluing intervals and adding the integrals of the corresponding indicator functions, and to prove that the image of this addition is a left $\\Lambda$-module. Theorem 3.14 states that the interval-forgetting restriction to this image is a $\\Lambda$-epimorphism onto $\\{I_\\Lambda\\}\\times k$, which is the categorical formulation of adding integrals with different upper limits. Theorem 3.17 then shows that under the L-conditions the additivity law holds for the corresponding numbers, and Corollary 4.3 identifies the i-poset space with the space of continuous functions vanishing at $c$, thereby recovering the known categorical derivation of the Lebesgue integral and showing the new object represents absolutely continuous functions.","pith_inferences":["The same initial-object-plus-addition recipe would work for any integral transform whose kernel defines a morphism from the initial step-function object, so a categorical addition formula for Laplace or Fourier transforms is a plausible next step, though the paper does not state this.","The trigonometric construction suggests a categorical route to elliptic functions: taking inverses of elliptic integrals with variable upper limits inside $A^p_\\Lambda$ could produce functions whose double periodicity and addition theorems might be derived from the same i-poset addition; this is an extension, not a claim of the paper.","The global-dimension formulas express a homological invariant as an integral of vertex homomorphisms; if the quoted equality $\\mathrm{gl.dim}\\,A=\\sup_F \\ell(F)$ holds for a wider class of quadratic monomial algebras, the identity $\\ell(q)=2\\int v_q$ would give integral characterizations beyond gentle algebras."],"forward_implications":["The classical additivity $\\int_\\alpha^\\beta f\\,d\\mu + \\int_\\beta^\\gamma f\\,d\\mu = \\int_\\alpha^\\gamma f\\,d\\mu$ is a direct corollary of the categorical addition structure (Theorem 3.17(1)).","The space of i-posets is R-linearly isomorphic to the Banach space of continuous functions on $[c,d]$ vanishing at $c$, recovering the known categorical derivation of the Lebesgue integral and giving a categorical model of absolutely continuous functions (Corollary 4.3).","Trigonometric functions arise as inverses of integrals with variable upper limit, and their period $2\\pi$ is computed from the circumference of a circle via a curvilinear integral in the same framework.","Logarithmic and exponential functions arise from the i-poset of $1/t$: $\\ln y_1+\\ln y_2=\\ln(y_1y_2)$ and $e^{x_1}e^{x_2}=e^{x_1+x_2}$ follow from the addition rule.","For gentle algebras with finite global dimension, $\\mathrm{gl.dim}\\,A/2$ is a supremum of multiple Lebesgue integrals over forbidden threads of the algebra and over permitted threads of its Koszul dual, and $\\mathrm{gl.dim}\\,A$ is a supremum of Lebesgue–Stieltjes integrals of the projections of permitted threads (Theorems 5.5 and 5.6)."],"supporting_citations":[{"why":"Supplies the original category $A^p$ and the object $(C_*([c,d]),\\mathrm{id},\\eta)$ that this paper recovers as Corollary 4.3, plus the L-conditions under which the canonical morphism becomes the Lebesgue integral.","marker":"[17]"},{"why":"Quoted for Theorems 2.16 and 2.17, the initiality of the completed step-function space and the Lebesgue-integral representation of its unique morphism, on which the main construction rests.","marker":"[21]"},{"why":"Quoted for the fact $\\mathrm{gl.dim}\\,A=\\sup_{F\\in\\mathrm{forb}(A)}\\ell(F)$, which carries the substance of the gentle-algebra applications in Theorems 5.5 and 5.6.","marker":"[20]"},{"why":"Supplies the change-of-variables and Lebesgue–Stieltjes integral identities used in Sections 5.2.1 and 5.5 to define logarithms and to convert the gentle-algebra sup into Stieltjes integrals.","marker":"[13]"},{"why":"Supplies the explicit description of the Koszul dual quiver and relations for $A^!$, used in the gentle-algebra application to write integrals over permitted threads.","marker":"[25]"}],"fun_headline_variants":["Categorifying integrals with variable endpoints","Variable-limit integrals become category morphisms","Integral posets unify Lebesgue additivity","A new categorical take on variable-limit integration","Category morphisms add variable-limit integrals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on previously established theorems, quoted here without proof, that the completed step-function space is initial in $A^p_\\Lambda$ and that its unique morphism to the scalar field is the Lebesgue integral; if either theorem is unsound, the epimorphism, the additivity law, and Corollary 4.3 all collapse.","fun_headline_variants_meta":{"raw":{"variants":["Categorifying integrals with variable endpoints","Variable-limit integrals become category morphisms","Integral posets unify Lebesgue additivity","A new categorical take on variable-limit integration","Category morphisms add variable-limit integrals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000515,"raw_usage":{"total_tokens":2509,"prompt_tokens":960,"completion_tokens":1549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":1490}},"tokens_in":576,"tokens_out":1549,"duration_ms":10299,"temperature":1.0,"reasoning_tokens":1490,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:46:32.337321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of Theorem 3.17(1) for $\\Lambda=\\mathbb{R}$, $c=0$, $d=3$, $\\alpha=0$, $\\beta=1$, $\\gamma=2$, and $f(x)=x$ using ordinary Lebesgue integrals; if the categorical sum prescribed by equation (3.3) does not equal $([0,2],3/2)$, the additivity theorem fails. Also check whether $k=\\mathbb{R}\\times[0,2\\pi)$ with multiplication $(r_1,\\theta_1)(r_2,\\theta_2)=(r_1r_2,\\theta_1+\\theta_2 \\bmod 2\\pi)$ is a field: the element $(0,\\pi/2)$ has no inverse, so the trigonometric setup of Section 5.1.2 does not satisfy the paper's normed-field axioms.","supporting_citations":[{"cited_title":"Normed modules and the categorification of integrations, series expansions, and differentiations","cited_arxiv_id":"2405.02777","evidence_quote":"Quoted for Theorems 2.16 and 2.17, the initiality of the completed step-function space and the Lebesgue-integral representation of its unique morphism, on which the main construction rests."},{"cited_title":"Mart ´ ınez-Villa","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit description of the Koszul dual quiver and relations for $A^!$, used in the gentle-algebra application to write integrals over permitted threads."}],"review_version":1}