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Normed representations of weight quivers

T0 review · 5 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single initial object in a category of normed bimodules over quiver tensor rings forces every integration and every series expansion to be the unique structure-preserving map out of it.

desk verdict New categorical scaffolding for normed bimodules over weight quivers, but the main morphism theorem is false without an unstated measure-affine condition, and the advertised analysis applications don't hold. read the letter →

arxiv 2507.06962 v2 pith:A5Q2RHMR submitted 2025-07-09 math.RT math.CTmath.FA

Yu-Zhe Liu This is my paper · ORCID
classification math.RTmath.CTmath.FA MSC 16G1046B9946M40
keywords weightquiverstensorringsnormedbimodulesBanachmodulesinitialobjectDaniellintegrationBochnercategorificationof
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 tries to establish that Daniell, Bochner, and Lebesgue integration, together with Stone–Weierstrass approximation, Taylor expansion, and Fourier expansion, are not separate constructions but one universal map. It builds a category whose objects are normed $(A,B)$-bimodules, where $A$ and $B$ are tensor rings attached to weight quivers, each object carrying a distinguished element $v$ and a juxtaposition operation $\delta$ that combines $2^{\dim A}$ inputs into one. The main theorem produces an initial object in this category: the completion of the elementary simple functions $I_A\to B$, with the constant function $1_{I_A}$ and the gluing map $\widehat{\gamma}_\xi$. For $p=1$ over a field extending $\mathbb{R}$, initiality forces a unique morphism $\widehat{T}$ to the target $(B,\mu_{I_A}(I_A)1_B,\mathcal{A})$, and on simple functions it is $\widehat{T}(\sum_i b_i1_{I_i})=\sum_i b_i\mu_{I_A}(I_i)$, which satisfies the axioms of an integral. If the paper is correct, the question of whether integration is uniquely defined has a positive, categorical answer: every integration in this setting is the unique structure-preserving map out of the same object.

What carries the argument

The load-bearing object is the completed bimodule $\widehat{S}_\zeta(I_A)$ of elementary simple functions $f:I_A\to B$, together with the constant function $1_{I_A}$ and the juxtaposition map $\widehat{\gamma}_\xi$ that cuts the cube $I_A$ into $2^{\dim A}$ sub-cubes and reassembles $2^{\dim A}$ functions into one function on the whole cube. Iterating $\gamma_\xi$ produces a tower $E_0\subseteq E_1\subseteq\cdots$ with $E_{u+1}\cong E_u^{\oplus_p 2^{\dim A}}$, and the completion is realized as the inductive limit $\widehat{S}_\zeta(I_A)\cong\varinjlim E_u$. The proof that this object is initial—existence of a morphism to every triple $(N,v,\delta)$ and uniqueness of that morphism—is what forces every integral and every expansion to be the same structure-preserving map.

What would settle it

Recompute the displayed integral in the paper's own Example 7.1: Theorem 5.1 forces $(\mathscr{A}^1_\zeta)\int_{[0,1]^{\times 11}} \varsigma|_{[0,1]^{\times 11}}\,d\mu_{I_A}=\frac12(1_B+a+b+c)+J$; if the direct evaluation of the simple-function formula gives any other element of $B$, the claimed uniqueness and form of $\widehat{T}$ are false.

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Extended reading notes

Core claim

The category $\mathscr{A}^p_\zeta$ has an initial object $(\widehat{S}_\zeta(I_A), 1_{I_A}, \widehat{\gamma}_\xi)$, where $\widehat{S}_\zeta(I_A)$ is the completion of the $(A,B)$-bimodule of elementary simple functions $f:I_A\to B$, and $\widehat{\gamma}_\xi$ is the completed juxtaposition map. The uncompleted bimodule $S_\zeta(I_A)$ is an $\mathscr{A}^p_\zeta$-initial object in the larger normed-module category $\mathscr{N}\mathrm{or}^p_\zeta$. For $p=1$, with $F$ an extension of $\mathbb{R}$ and $A,B$ complete, there is a unique morphism $\widehat{T}$ from this initial object to $(B,\mu_{I_A}(I_A)1_B,\mathcal{A})$, given on elementary simple functions by $\widehat{T}(\sum_i b_i1_{I_i})=\sum_i b_i\mu_{I_A}(I_i)$; this morphism satisfies the Daniell-type axioms (I1), (I2), and (I3), so it is denoted $(\mathscr{A}^1_\zeta)\int_{I_A}(\cdot)\,d\mu_{I_A}$. Bochner and Lebesgue integrals are the same morphism with special choices of $A,B,\varsigma$, and the Stone–Weierstrass, Taylor, and Fourier results are the same initiality statement applied to submodules closed under juxtaposition.

Load-bearing premise

The construction only works if the base field carries an ordered interval that can be split into two order-isomorphic pieces and measured, and the main integration theorems then also need the field to extend the real numbers and the algebras to be complete.

Editorial extensions

If this is right

  • If the main theorem is correct, for fixed data $(A,B,\varsigma,\mu_{I_A})$ the integral of every integrable function in $\widehat{S}_\zeta(I_A)$ is already determined: the unique morphism $\widehat{T}$ computes it, so there is no freedom in defining integration on these bimodules.
  • The classical Lebesgue integral on $L^1([0,1])$ is the special case $A=B=F=\mathbb{R}$, $\varsigma=\mathrm{id}$, $I_A=[0,1]$, and the usual juxtaposition map, so the categorical description includes the standard theory as the same initial-object morphism.
  • Bochner integration of vector-valued functions is recovered when $A=\mathbb{R}^{d_A}$, $B=\mathbb{R}^{d_B}$, and $\varsigma=0$, identifying the universal morphism with the Bochner integral.
  • Stone–Weierstrass becomes the statement that the completion of any juxtaposition-closed submodule has exactly one morphism from the initial object; Taylor and Fourier series are the two explicit instances constructed in Sections 6.2 and 6.3.
  • The uniqueness result answers the paper's Question 1.1(Q3): among normable $(A,B)$-bimodules of this type, there is exactly one integration theory once the data are fixed.

Reading between the lines

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

  • Beyond the paper: the framework suggests that an integral is determined by its values on indicator functions, since the initial object is built from elementary simple functions; this is a categorical form of the usual fact that a measure determines an integral.
  • Beyond the paper: the proof of $(A,B)$-linearity of $\widehat{T}$ uses the fact that $\mu_{I_A}(I_i)\in\mathbb{R}$ is fixed by all Galois automorphisms of the tensor-ring modulation; a natural testable extension is to ask whether the uniqueness theorem survives for ordered fields not extending $\mathbb{R}$ if one replaces commutativity of the measures by a centrality condition.
  • Beyond the paper: the same initial-object construction might apply to other averaging and expansion operations beyond Taylor and Fourier, such as wavelet or interpolation schemes, whenever the target admits a suitable juxtaposition map $\delta$.
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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

5 major / 4 minor

Summary. The paper introduces normed tensor rings and (A,B)-bimodules, defines categories Nor^p_ζ and A^p_ζ whose objects are triples (N,v,δ), and claims that A^p_ζ has an initial object constructed from elementary simple functions on I_A. It further claims that the unique morphism from this initial object to a target object (B, μ_{I_A}(I_A)1_B, A) recovers Daniell, Bochner, and Lebesgue integration, and that the same initial-object formalism categorifies the Stone–Weierstrass theorem, power series expansion, and Fourier expansion. The central technical statements are Theorem 4.18 (initial object), Theorem 4.21 (Nor^p_ζ-initial object), and Theorem 5.1 (unique integral morphism).

Significance. The categorical unification attempted here is an interesting extension of Leinster's categorification of Lebesgue integration to general weight-quiver tensor rings and bimodules, and the paper contains explicit constructions and a worked example in Section 7. If the initial-object theorem were correct, the uniqueness statements would indeed give a clean answer to Question 1.1(Q3). However, the main morphism property (H2) in Theorem 5.1 is false under the stated assumptions, and a key density claim in Section 6.2 is also false. These are not local presentation issues but load-bearing errors in the central claims.

major comments (5)
  1. [Section 5.1, proof of Theorem 5.1(1)] The verification of (H2) is incorrect. In the third displayed chain of the proof, the equality A(\tilde T^{⊕2^{d_A}}((f_t))) = \tilde T(γ_ξ((f_t))) is asserted, but the computation replaces μ(κ_t(I_i)) by (μ(I_t)/μ(I_A))μ(I_i). This replacement is valid only if each juxtaposition bijection κ_t scales Lebesgue measure by a constant factor, a condition that is nowhere stated or proved. Concrete counterexample: take d_A=1, F=R, A=B=R, I_A=[0,1], ξ=1/2, μ=Lebesgue, and let κ_c(t)=t^2/2, κ_d(t)=1/2+t^2/2, which are allowed by the standing assumption before Definition 4.3. Set f_1=1_{[0,1/2]} and f_2=0. Then γ_ξ(f_1,f_2)=1_{κ_c([0,1/2])}=1_{[0,1/8]}, so T(γ_ξ(f_1,f_2))=1/8, while A(Tf_1,Tf_2)=(1/2)(1/2)+(1/2)(0)=1/4. Thus (H2), Theorem 5.1, and the claimed categorical description of Daniell/Bochner/Lebesgue integration fail as stated.
  2. [Lemma 4.1 and Notation 4.2] Lemma 4.1 has an index mismatch: it states a disjoint union I_A=∪_{i=1}^{d_A} I_i, but the norm being defined is over 2^{d_A} direct summands X_i. The proof then assumes μ(I_i)/μ(I_A)=c for all i with c d_A=1, which is not part of the hypothesis and is generally false. Although the conclusion (N2) can in fact be verified without the constant assumption, the proof as written is invalid, and the same d_A versus 2^{d_A} confusion reappears in the H2 computation in Theorem 5.1. This needs to be repaired because the norm on N^{⊕_p 2^{d_A}} is used throughout the construction of the category and the initial object.
  3. [Proposition 4.16, proof of Theorem 4.18] The boundedness of the constructed morphism is not established. After defining θ_u recursively, the proof asserts that ∥θ_1∥=sup_{...}=∥δ∥ and then that ∥θ_t∥=∥δ∥ for all t by induction. The displayed equality for ∥θ_1∥ is not justified: the sup over the constraint involving μ(I_i) is not shown to equal ∥δ∥, and no uniform bound on the family θ_u is derived. Since the existence of a bounded limit θ_lim is used to justify the passage from (4.4) to the equality \tilde θ \hat γ_ξ = δ \tilde θ^{⊕2^{d_A}}, the proof of Theorem 4.18 is incomplete.
  4. [Section 6.2, Eq. (6.1)] The statement that R[x,x^{-1}] is dense in L^1([0,1]) is false: the functions x^{-n} for n≥1 are not in L^1([0,1]), so the subspace is not even contained in L^1([0,1]). Consequently the asserted isomorphism \widehat{R[x,x^{-1}]} ≅ L^1([0,1]) in (6.1) is false, and the claimed unique morphism H_pow categorifying Taylor expansion is not established. The Taylor map as written lands in R[x], not R[x,x^{-1}], which further indicates that the density claim is not merely a typo.
  5. [Abstract and Section 4] The abstract and Question 1.1 advertise results for arbitrary tensor rings over a field F, but from Section 4 (before Definition 4.3) onward the paper assumes that F contains a totally ordered interval [c,d]_F with a measure μ_F and order-preserving bijections κ_c,κ_d; Theorem 5.1 additionally requires F to be an extension of R. These are substantial restrictions on the advertised generality and should be stated prominently in the abstract and introduction.
minor comments (4)
  1. [Definition 4.6 and Theorem 5.1] Definition 4.6 defines elementary simple functions with coefficients k_i∈F, but Theorem 5.1 writes f=Σ b_i 1_{I_i} with b_i∈B; the relationship between the F-valued coefficients and the B-valued ones (via ς or otherwise) should be clarified.
  2. [Section 4.3.1, Definition 4.3(N2)] The notation P:B×I→N with P((1_B)_{1×I})=v is confusing: B×I is used as a Cartesian product, but the proof later treats (1_B)_{1×I} as a family indexed by I. The intended construction should be described more explicitly.
  3. [Section 2.2] There is a duplicated phrase in the text: "It follows that each M can be It follows that each M can be" should read "It follows that each M can be". Similar typographical slips appear elsewhere, including inconsistent use of "completed" for "complete".
  4. [Section 6.1, Corollary 6.2] Corollary 6.2 is a direct consequence of the initial-object uniqueness and does not by itself prove any approximation statement; the name "Stone–Weierstrass Approximation Theorem" and the surrounding discussion should be adjusted so as not to suggest that a categorical uniqueness statement is an approximation theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the initial-object theorem is proved from the category axioms, and the integral morphism is verified rather than assumed.

full rationale

The central claim is Theorem 4.18, and its proof is self-contained: Proposition 4.16 constructs a morphism from (hat S_zeta(I_A), 1, hat gamma_xi) to an arbitrary object (N,v,delta) by recursion over the filtration E_u, while Proposition 4.17 proves uniqueness from (H1) and (H2) by induction. Neither step uses the integral formula that Theorem 5.1 later states. Theorem 5.1 proceeds by the standard universal-property method: it proposes the candidate T~(sum b_i 1_{I_i}) = sum b_i mu_{I_A}(I_i) and verifies (H1), (H2), and the (A,B)-module axioms; the candidate is not obtained by fitting any parameter to the target data, and the verification does not reduce to the defining equation of T~. The citations to the author's earlier work [31,32] are used for background definitions and for the scalar case of the normed-module framework, but the (A,B)-bimodule initial-object theorem and the uniqueness argument are proved in this paper from the category axioms. No load-bearing uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed as a prediction. A separate correctness concern, not a circularity, is that the displayed verification of (H2) in Theorem 5.1 appears to require the order-preserving bijections kappa_c and kappa_d to multiply the measure by the constant mu(I_t)/mu(I_A); without that stated condition the displayed chain is algebraically suspect. That issue does not make the derivation circular.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper imports a heavy setup: finite-dimensional tensor rings from weight-quiver modulations, a chosen zeta: A to B, an ordered interval inside F, a measure mu_F, norm basis functions, and completeness assumptions. The free choices p, xi, mu, and norm basis functions are not fitted to data but control the results. The power-series section adds a false density axiom. No new physical entities such as particles or forces are introduced; the triples and juxtaposition maps are mathematical definitions.

free parameters (5)
  • p
    The exponent 1 <= p in R appears in the norms (3.1) and (3.2) and in the category A^p_zeta; the integration theorem is proved only for p=1.
  • xi = any interior point of the interval
    The juxtaposition map gamma_xi cuts I_A at xi; different xi give different initial objects in the same category.
  • mu_F and mu_{I_A} = arbitrary measure
    The integral map is sum b_i mu(I_i); changing the measure changes the integral. mu is an input, not derived.
  • norm basis functions n_A, n_B
    The norms on A and B depend on arbitrary choices n: basis to R_{>=0}; these choices affect the normed module category.
  • ordered interval [c,d]_F
    The framework assumes F contains a totally ordered interval [c,d]_F with a measure; this restricts the base field and is not forced by the quiver data.
assumptions (5)
  • ad hoc to paper F contains a totally ordered subset I=[c,d]_F with a measure mu_F and order-preserving bijections dividing I at xi.
    Introduced in Section 4 before Definition 4.3 to define I_A and gamma_xi; not available for arbitrary fields and not stated in the abstract.
  • domain assumption A and B are finite-dimensional tensor rings given by weight quivers with chosen modulations, and zeta: A to B is a homomorphism.
    Used throughout Sections 2 and 3; this is the intended domain of the paper.
  • domain assumption F is an extension of R and F, A, B are complete in Theorems 1.3, 1.4, and 5.1.
    Assumed at the start of Section 4.4.2 and in Theorem 5.1; the abstract presents the results without these restrictions.
  • standard math Standard measure-theoretic facts hold for products of intervals: mu_{I_A}(I_A) = mu_F([c,d])^{d_A} and additivity over disjoint unions.
    Used to compute norms and integrals in Section 4 and Theorem 5.1.
  • ad hoc to paper R[x,x^{-1}] is dense in L^1([0,1]) and its completion is L^1([0,1]).
    Used in Section 6.2, equation (6.1). This axiom is false because x^{-1} is not integrable on [0,1], so the power-series application is unsupported.

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Cite this review

Pith. "Pith review of Normed representations of weight quivers." pith.science (2026). https://pith.science/paper/A5Q2RHMR

@misc{pith2026250706962,
  author       = {Pith},
  title        = {Pith review of: Normed representations of weight quivers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A5Q2RHMR}},
  note         = {Machine review of arXiv:2507.06962}
}
abstract

Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\mathscr{A}^p_{\varsigma}$ in this paper whose object is a triple $(N,v,\delta)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\delta: N^{\oplus_p 2^{\dim A}} \to N$ and each morphism $(N,v,\delta) \to (N',v',\delta')$ is given by an $(A,B)$-homomorphism $\theta: N\to N'$ such that $\theta(v)=v'$ and $\delta' \theta^{\oplus 2^{\dim A}} = \theta\delta$ hold. We show that $\mathscr{A}^p_{\varsigma}$ has an initial object such that Daniell integration, Bochner integration, Lebesgue integration, Stone--Weierstrass Approximation Theorem, power series expansion, and Fourier series expansion are morphisms in $\mathscr{A}^p_{\varsigma}$ starting with this initial object.

Figures

Figures reproduced from arXiv: 2507.06962 by the authors.

Figure 4.1
Figure 4.1. Juxtaposition map and for each u ∈ N, we define Eu+1 := Im(γξ|Eu ). Then we have the following lemma. Lemma 4.11. All Eu are normed (A, B)-bimodules. Furthermore, for each u ∈ N, γξ|Eu provide an isomorphism E ⊕p2 dA u ∼= Eu+1 between two (A, B)-bimodules. Proof. For any u ∈ N, one can check that Eu is an (A, B)-bimodule in a way similar to the proof of Lemma 4.9. By the definition of Eu, it is clear that γξ|Eu is a… view at source ↗
Figure 4.2
Figure 4.2. The existence of (A, B)-homomorphism (S\ς(IA), 1IA , γbξ) → (N, v, δ). Thus, naturally, we need to consider the following diagram up to the isomorphism ρ: E ⊕ ui γξ|E ⊕ ui ∼= /  _ e ⊕2 dA ui  θ ⊕2 dA ui & Eui+1 _ eui+1  θui x Sb⊕ γbξ / θ˜⊕2 dA  S\ς(IA) θ˜  N ⊕ δ /N, where, for each t ∈ N, et := ραt is an embedding. We have the following equation ˜θ(γbξ(f )) = lim←− ˜θ(γbξ(e ⊕2 dA ui (f i))) = lim←− ˜θ(eui+… view at source ↗
Figure 4.3
Figure 4.3. The uniqueness of (A, B)-homomorphism (S\ς(IA), 1IA , γbξ) → (N, v, δ). it follows h|E0 = h ′ |E0 . Therefore, one can prove that h|Eu = h ′ |Eu for all u ∈ N by induction. The direct system (Ei)i∈N,(eij : Ei ⊆→ Ej )i⩽j  provides a commutative diagram shown in [PITH_FULL_IMAGE:figures/full_fig_p032_4_3.png] view at source ↗
Figures from the paper (2 more)
Figure 7.1
Figure 7.1. Figure 7.1: Quiver QA 0 44 [PITH_FULL_IMAGE:figures/full_fig_p044_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: Quiver QA α ∈ (QA)1, we have Aα = R ⊗R∩R R ∼= R. Next, let B = lkQB/IB be given by the quiver QB := Q and the admissible ideal IB = ⟨ab, bc, ca⟩. Then A/J (∼= B) induced an epimorphism ς : A → B, x 7→ x + J , where J = ⟨x1 + IA, x2 + IA, a′ + IA, b′ + IA, c′ + IA⟩. C…

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Works this paper leans on

44 extracted references · 29 canonical work pages

  1. [1]

    Alvarez-Picallo and J.-S

    M. Alvarez-Picallo and J.-S. P. Lemay. Cartesian difference categories.Log. Methods Comput. Sci., 17(3):paper no. 23, 48, 2021. DOI:10.46298/lmcs-17(3:23)2021

    doi:10.46298/lmcs-17(3:23)2021 2021

  2. [2]

    Assem, D

    I. Assem, D. Simson, and A. Skowro´ nski.Elements of the Representation Theory of Associative Algebras, Volume 1 Techniques of Representation Theory. Cambridge University Press, Cambridge, 2006. DOI:10.1017/CBO9780511614309

    doi:10.1017/cbo9780511614309 2006

  3. [3]

    G. E. Baxter. An analytic problem whose solution follows from a simple algebraic identity.Pacific J. Math., 10(3):731–742, 1960. DOI:10.2140/pjm.1960.10.731. 0 46 Y.-Z. LIU: Normed representations of weight quivers

    doi:10.2140/pjm.1960.10.731 1960

  4. [4]

    Bennett-Tennenhaus and W

    R. Bennett-Tennenhaus and W. Crawley-Boevey. Semilinear clannish algebras.Proc. Lond. Math. Soc., 129(4): paper no. e12637, 2024. DOI:10.1112/plms.12637

    work page doi:10.1112/plms.12637 2024

  5. [5]

    Semilinear clannish algebras arising from surfaces with orbifold points

    R. Bennett-Tennenhaus and D. Labardini-Fragoso. Semilinear clannish algebras aris- ing from surfaces with orbifold points. arXiv:2303.05326, 2023

    work page Pith review arXiv 2023

  6. [6]

    R. F. Blute, J. R. B. Cockett, and R. A. G. Seely. Differential categories.Math. Struct. Comput. Sci., 16(06):1049–1083, 2006. DOI:10.1017/S0960129506005676

    doi:10.1017/s0960129506005676 2006

  7. [7]

    R. F. Blute, J. R. B. Cockett, and R. A. G. Seely. Cartesian differential storage categories.Theor. Appl. Categ., 30:620–687, 2015. AMS:MR3346979

    work page 2015

  8. [8]

    S. Bochner. Integration von funktionen, deren werte die elemente eines vektorraumes sind.Fund. Math., 20(1):262–176, 1933. https://bibliotekanauki.pl/articles/1385181

    work page arXiv 1933

Show all 44 references
  1. [9]

    J. R. B. Cockett and J.-S. P. Lemay. Cartesian integral categories and contextual integral categories. InProceedings of the Thirty-Fourth Conference on the Mathemat- ical Foundations of Programming Semantics (MFPS XXXIV), volume 341 ofElec- tron. Notes Theor. Comput. Sci., pag...

    2018 doi

  2. [10]

    J. R. B. Cockett and J.-S. P. Lemay. Integral categories and calculus categories.Math. Struct. Comp. Sci., 29(2):243–308, 2018. DOI:10.1017/S0960129518000014

    2018 doi

  3. [11]

    P. J. Daniell. A general form of integral.Ann. Math., 19(4):279–294, 1918. DOI:10.2307/1967495

    1918 doi

  4. [12]

    P. J. Daniell. Functions of limited variation in an infinite number of dimension.Ann. Math., 21(1):30–38, 1919. DOI:10.2307/2007133

    1919 doi

  5. [13]

    P. J. Daniell. Integrals in an infinite number of dimensions.Ann. Math., 20(4):281– 288, 1919. DOI:10.2307/1967122

    1919 doi

  6. [14]

    P. J. Daniell. Two generalizations of the stieltjes integral.Ann. Math., 23(2):169–182,

  7. [15]

    Dlab and C

    V. Dlab and C. M. Ringel. On algebras of finite representation type.J. Algebra, 33(2):306–394, 1975. DOI:10.1016/0021-8693(75)90125-8

    1975 doi

  8. [16]

    P. Gabriel. Indecomposable representations I.Manuscripta Math., 6:71–109, 1972. 0 47 2025-7-14

    1972

  9. [17]

    P. Gabriel. Indecomposable representations II.Symposia Math., 11:81–104, 1973

    1973

  10. [18]

    P. Gabriel. Repr´ esentations ind´ ecomposables. InS´ eminaire Bourbaki vol. 1973/74 Expos´ es 436–452, pages 143–169. Springer, 2006. DOI:10.1007/BFb0066369

    1973 doi

  11. [19]

    Geuenich.Quiver modulations and potentials

    J. Geuenich.Quiver modulations and potentials. PhD thesis, Universit¨ ats-und Lan- desbibliothek Bonn, 2017.https://hdl.handle.net/20.500.11811/7162

    2017

  12. [20]

    Geuenich and D

    J. Geuenich and D. Labardini-Fragoso. Species with potential arising from surfaces with orbifold points of order 2, part I: one choice of weights.Math. Z., 286(3):1065– 1143, 2016. DOI:10.1007/s00209-016-1795-6

    2016 doi

  13. [21]

    Geuenich and D

    J. Geuenich and D. Labardini-Fragoso. Species with potential arising from surfaces with orbifold points of order 2, part II: Arbitrary weights.Int. Math. Res. Notices, 2020(12):3649–3752, 2020. DOI:10.1093/imrn/rny090

    2020 doi

  14. [22]

    Gigli and E

    N. Gigli and E. Pasqualetto. The theory of normed modules. InLectures on Nonsmooth Differential Geometry, SISSA Springer Series (SISSASS), pages 67–69, Cham, 2020. Springer. DOI:10.1007/978-3-030-38613-9 3

    2020 doi

  15. [23]

    E. L. Green. The representation theory of tensor algebras.J. Algebra, 34(1):136–171,

  16. [24]

    T. Guo. On some basic theorems of continuous module homomorphisms be- tween random normed modules.J. Funct. Space., 54(1):115–130, 2013. DOI: 10.1155/2013/989102

    2013 doi

  17. [25]

    Ikonicoff and J.-S

    S. Ikonicoff and J.-S. P. Lemay. Cartesian differential comonads and new models of Cartesian differential categories.Cah. Topol. G´ eom. Diff´ er. Cat´ eg., 64(2):198–239,

  18. [26]

    Labardini-Fragoso and A

    D. Labardini-Fragoso and A. Zelevinsky. Strongly primitive species with poten- tials I: mutations.Bolet ´ ın de la Sociedad Matem´ atica Mexicana, 22(1):47–115, 2016. DOI:10.1007/s40590-015-0063-9

    2016 doi

  19. [27]

    H. L. Lebesgue.Le¸ cons sur l’int´ egration et la recherche des fonctions primitives: professees au college de France (deuxiemeedition). Collection de monographies sur la theorie des fonctions, 1928. 0 48 Y.-Z. LIU: Normed representations of weight quivers

    1928

  20. [28]

    Leinster

    T. Leinster. A categorical derivation of lebesgue integration.J. Lond. Math. Soc., 107(6):1959–1982, 2023. DOI:10.1112/jlms.12730

    1959 doi

  21. [29]

    J.-S. P. Lemay. Differential algebras in codifferential categories.J. Pure. Appl. Algebra, 223(10):4191–4225, 2019. DOI:10.1016/j.jpaa.2019.01.005

    2019 doi

  22. [30]

    J.-S. P. Lemay. Cartesian Differential Kleisli Categories.Electronic Notes in Theoret- ical Informatics and Computer Science, 3(23), 2023. DOI:10.46298/entics.12278

    2023 doi

  23. [31]

    M. Liu, S. Liu, and Y.-Z. Liu. Normed modules, integral sequences, and integrals with variable upper limits.B. Iran. Math. Soc., 0(0):1–45, 2025. DOI:10.1007/s41980-025- 00990-4, arXiv:2411.19904

    2025 arXiv

  24. [32]

    Y.-Z. Liu, S. Liu, Z. Huang, and P. Zhou. Normed modules and the categorification of integrations, series expansions, and differentiations.Sci. China Math., pages paper no. SCM–2024–0654, 2025. DOI:10.1007/s11425-024-2418-3

    2024 doi

  25. [33]

    C. M. Ringel. Representations ofK-species and bimodules.J. Algebra, 41(2):269–302,

  26. [34]

    G.-C. Rota. Baxter algebras and combinatorial identities I.Bull. Amer. Math. Soc., 75:325–329, 1969. MR0244070, zbMATH:0192.33801

    1969

  27. [35]

    G.-C. Rota. Baxter algebras and combinatorial identities II.Bull. Amer. Math. Soc., 75:330–334, 1969. MR0244070. zbMATH:0319.05008

    1969

  28. [36]

    J. J. Rotman.An Introduction to Homological Algebra (Second Edition). Academic Press, 1979. DOI:10.1007/b98977

    1979 doi

  29. [37]

    H. L. Royden. The Daniell integral. InReal Analysis, pages 419–434. Prentice Hall, Englewood Cliffs, 3 edition, 1988. ISBN:0-02-404151-3

    1988

  30. [38]

    I. Segal. Algebraic integration theory.Bull. Amer. Math. Soc., 71(3):419–489, 1965. DOI:S0002-9904-1965-11284-8

    1965

  31. [39]

    D. Simson. Categories of representations of species.J. Pure Appl. Algebra, 14(1):104– 114, 1979. DOI:10.1016/0022-4049(79)90015-X

    1979 doi

  32. [40]

    M. H. Stone. The generalized Weierstrass approximation theorem.Mathematics Mag- azine, 21(5):237–254, 1948. DOI:10.2307/3029337. 0 49 2025-7-14

    1948 doi

  33. [41]

    Weierstrass

    K. Weierstrass. ¨Uber die analytische darstellbarkeit sogenannter willk¨ urlicher func- tionen reeller argumente.Sitzungsberichte der K¨ oniglich Preußischen Akademie der Wissenschaften zu Berlin, pages 633–639, 1885.https://archive.org/details/ sitzungsberichte1885deutsch/pag...

  34. [42]

    Weierstrass

    K. Weierstrass. ¨Uber die analytische darstellbarkeit sogenannter willk¨ urlicher functio- nen reeller argumente. In J. Knoblauch, editor,Mathematische Werke: Herausgegeben unter Mitwirkung einer von der k¨ oniglich preussischen Akademie der Wissenschaften eingesetzten Commiss...

    2013 doi

  35. [1975]

    DOI:10.1016/0021-8693(75)90200-8

    doi

  36. [1976]

    DOI:10.1016/0021-8693(76)90184-8

    doi

Pith tools

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