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arxiv: 2403.01067 · v2 · submitted 2024-03-02 · 🧮 math.AT · math-ph· math.GT· math.MP· math.QA

Nested cobordisms, Cyl-objects and Temperley-Lieb algebras

Maxine E. Calle , Renee S. Hoekzema , Laura Murray , Natalia Pacheco-Tallaj , Carmen Rovi , Shruthi Sridhar-Shapiro This is my paper

Pith reviewed 2026-05-24 02:57 UTC · model grok-4.3

classification 🧮 math.AT math-phmath.GTmath.MPmath.QA
keywords nested cobordismsstriped cylinder categoryTemperley-Lieb algebrascyclic objectsCyl-objectscobordism categoriesmonoidal categories
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The pith

The striped cylinder cobordism category Cyl has a complete presentation by generators and relations, with its objects in a category C corresponding to Temperley-Lieb algebras and cyclic objects.

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

The paper defines a category of nested cobordisms between nested manifolds using a discrete model. It applies a variation of stratified Morse theory to obtain generators for the general case and then restricts to the striped cylinder category Cyl to find all relations among those generators. Functors from Cyl to another category C are shown to correspond to structures already studied in algebra, and two new constructions are introduced that mimic features of the cobordism category.

Core claim

Restricting to the striped cylinder cobordism category Cyl yields a complete set of relations for the generators obtained from stratified Morse theory. Cyl-objects, defined as functors from Cyl to a target category C, relate directly to Temperley-Lieb algebras and to cyclic objects. The structure also inspires a doubling construction on cyclic objects and a cylindrical bar construction on self-dual objects in a monoidal category.

What carries the argument

The striped cylinder cobordism category Cyl, whose generators and relations provide the presentation that links cobordism functors to algebraic structures such as Temperley-Lieb algebras.

Load-bearing premise

A variation of stratified Morse theory applies to the nested case and produces the correct generators for the general nested cobordism category.

What would settle it

An explicit check that a proposed relation among generators in Cyl cannot be derived from the listed relations, or a Cyl-object that does not match any Temperley-Lieb algebra structure.

Figures

Figures reproduced from arXiv: 2403.01067 by Carmen Rovi, Laura Murray, Maxine E. Calle, Natalia Pacheco-Tallaj, Renee S. Hoekzema, Shruthi Sridhar-Shapiro.

Figure 1
Figure 1. Figure 1: Generating cobordisms A complete description of the relations is given in Theorem 3.14, which includes the usual relations on 1-dimensional cobordisms (the snake relation, etc.) as well as relations involving how the twist interacts with the birth and death cobordisms. The motivation for providing a generators and relations description of Cyl (and Cob1<2 more generally) is to understand the explicit data n… view at source ↗
Figure 2
Figure 2. Figure 2: An example of a nested manifold M0<1<2: a surface endowed with a 1- dimensional submanifold, which is itself endowed with a 0-dimensional submanifold. Nested manifolds are a special case of stratified manifolds, see Lemma 2.17. In particu￾lar, stratified manifolds allow for singularities at substrata that are not allowed for nested manifolds; see [GM88] for more background on stratified manifolds. Remark 2… view at source ↗
Figure 3
Figure 3. Figure 3: The figure on the left is an example of an individually Morse function which is not nested Morse. The figure on the right is nested Morse. We claim that every nested function can be approximated by a nested Morse function (Theorem 2.15). Lemma 2.14. [See [ADE14], Proposition 1.2.1] Given an embedding of MI into R N , we have that for almost all p ∈ R N , the function fp : MI → R, x 7→ ||x − p||2 is a neste… view at source ↗
Figure 4
Figure 4. Figure 4: Example of Morse data for the point p. We have defined a nested Morse function as individually Morse with critical points and values being distinct. Since the critical values of a nested Morse function are isolated, it suffices to look at Morse data for a small neighborhood around a critical point p; we describe this construction of local Morse data. Provide Mdn with a smooth Riemannian metric. [GM88] show… view at source ↗
Figure 5
Figure 5. Figure 5: Example of normal and tangential Morse data for a local picture with a critical point of index 1 in the submanifold M1 and no critical points on M2; N is the normal slice. Remark 2.25. In the situation of a nested manifold, the normal and tangential Morse data for a critical point on stratum Fi take a particularly nice form. For Whitney stratified spaces, the topological type of the boundary of the normal … view at source ↗
Figure 6
Figure 6. Figure 6: The points in red mark examples of each type of critical point that a nested Morse function M1<2 → R can have. Note that these pictures do not just show the local neighborhood of the indicated points but also include other critical points of different types. 2.3. Nested Cerf decompositions. Using the nested Morse theory of the previous section, we now outline how any nested cobordism can be written as a co… view at source ↗
Figure 7
Figure 7. Figure 7: Generating cobordisms Theorem 3.12. The elementary cobordisms in Definition 3.10 generate all morphisms in Cyl. Proof. By Corollary 2.36, every morphism in Cylc can be written as a composition of ele￾mentary cobordisms. So it suffices to show that the list from Definition 3.10 generates all elementary cobordisms in Cylc , which will then also provide a complete list of generators for the quotient category … view at source ↗
Figure 8
Figure 8. Figure 8: Relations involving interactions between birth and deaths Remark 3.15. The deaths (and births) that cannot be moved past each other are ‘stacked’ (for example: d j−1 k−2 ◦ d j k for 1 ≤ j < k − 2). We will show that the relations in 3.14 are sufficient, but our argument proceeds by putting every cobordism in a normal form. That process is easier to describe by knowing the full set of pairs of births and/or… view at source ↗
Figure 9
Figure 9. Figure 9: Relations involving birth or deaths moving past each other = = i i i j j-2 j j+2 i = = = i i i +1 i +1 (a) Birth - Twist commute = = i i i j j-2 j j+2 i = = = i i i +1 i +1 (b) Death - Twist commute = = i i i j j-2 j j+2 i = = = i i i +1 i +1 (c) Dehn twist [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Relations involving twists Corollary 3.16. The following “edge case” relations hold in Cyl: • Relations where births and deaths interact: (0∗ ) bracelet: d 1 2 ◦ b 0 0 = d 0 2 ◦ b 1 0 (2∗ ) untwisted snakes: d k k+2 ◦ b k+1 k = idk and d k+1 k+2 ◦ b k k = idk (2∗∗) twisted snakes: d k+1 k+2 ◦ b 0 k = tw2 k and d 0 k+2 ◦ b k+1 k = twk−2 k (3∗ ) no ‘interaction’ between birth and death: d i k+2 ◦ b k+1 k = … view at source ↗
Figure 11
Figure 11. Figure 11: bracelet relation • We can move deaths past each other as long as they are not ‘stacked.’ The ‘stacked’ deaths are d j−1 k−2 ◦ d j k for 1 ≤ j < k − 2, d i k−2 ◦ d j k if (i, j) = (0, k − 1),(k − 3, 0) or (k − 3, k − 2). • We can move deaths before births except when they are at the same spot (creating contractible circles that we impose to be the identity) or are adjacent. • When births and deaths are ad… view at source ↗
Figure 12
Figure 12. Figure 12: Example of a tangle. Two annular tangles are said to be equivalent if there is a orientation-preserving diffeomor￾phism between the two. Composition of tangles is given by nesting annuli, after isotoping the strings to line up the marked regions [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The composition of tangles in Atl [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Example of the map Λ ,→ Cyl. Definition 4.18. Let √ Λ op be the category with the same generators and relations as Λop (Definition 4.15) except that that tn is replaced with a generator called √ tn : [n] → [n], and relations (iv)–(vi) are replaced by the following: (iv) √ tn 2(n+1) = id, (v) √ tn+1 2 ◦ s j n = s j+1 n ◦ √ tn 2 , (vi) √ tn 2 ◦ d i n+1 = d i+1 n+1 ◦ √ tn+1 2 . The inclusion Λop → √ Λ op is … view at source ↗
read the original abstract

We introduce a discrete cobordism category for nested manifolds and nested cobordisms between them. A variation of stratified Morse theory applies in this case, and yields generators for a general nested cobordism category. Restricting to a low-dimensional example of the ``striped cylinder'' cobordism category Cyl, we give a complete set of relations for the generators. With an eye towards the study of TQFTs defined on a nested cobordism category, we describe functors Cyl$\to\mathcal{C}$, which we call Cyl-objects in $\mathcal{C}$, and show that they are related to known algebraic structures such as Temperley-Lieb algebras and cyclic objects. We moreover define novel algebraic constructions inspired by the structure of Cyl-objects, namely a doubling construction on cyclic objects analogous to edgewise subdivision, and a cylindrical bar construction on self-dual objects in a monoidal category.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper introduces a discrete cobordism category for nested manifolds and nested cobordisms. It asserts that a variation of stratified Morse theory yields generators for the general nested cobordism category. Restricting to the striped cylinder category Cyl, it claims a complete set of relations among those generators. It defines Cyl-objects in a category C as functors from Cyl, relates them to Temperley-Lieb algebras and cyclic objects, and introduces a doubling construction on cyclic objects and a cylindrical bar construction on self-dual objects in monoidal categories.

Significance. If the generators and relations are correctly identified and the functors well-defined, the work supplies an algebraic presentation of a nested cobordism category that could support new TQFT constructions, while the doubling and bar constructions provide concrete algebraic tools extending known structures such as edgewise subdivision and bar constructions.

major comments (1)
  1. [Abstract] Abstract: The completeness of the relation set for the Cyl category is asserted relative to generators obtained from a variation of stratified Morse theory in the nested setting. No explicit statement of the stratification conditions, nesting depth restrictions, or verification that all critical loci and handle attachments are captured appears in the provided text; without this, the claim that the listed relations are complete cannot be assessed and is load-bearing for the central algebraic presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this point about the abstract. We address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The completeness of the relation set for the Cyl category is asserted relative to generators obtained from a variation of stratified Morse theory in the nested setting. No explicit statement of the stratification conditions, nesting depth restrictions, or verification that all critical loci and handle attachments are captured appears in the provided text; without this, the claim that the listed relations are complete cannot be assessed and is load-bearing for the central algebraic presentation.

    Authors: The abstract is a high-level summary and therefore omits the technical details of the stratified Morse theory variation, which are developed at length in the body of the paper (in the sections deriving the generators for the general nested cobordism category). We agree, however, that the abstract's brevity makes the completeness claim difficult to assess on its own. We will therefore revise the abstract to include a concise statement of the stratification conditions, the nesting-depth restrictions, and a brief indication that the variation captures the relevant critical loci and handle attachments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces the nested cobordism category, states that a variation of stratified Morse theory yields its generators, restricts to the Cyl subcategory to list explicit relations among those generators, defines Cyl-objects as functors to a target category C, and introduces independent algebraic constructions (doubling on cyclic objects, cylindrical bar construction). None of these steps reduce a claimed output to an input by definition, by fitting a parameter to related data, or by a self-citation chain whose content is itself unverified. The relations are presented as complete relative to the stated generators; the algebraic objects and constructions are defined directly rather than derived tautologically from the geometric data. The derivation chain therefore contains no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard category axioms and a domain-specific assumption from Morse theory; no numerical free parameters or newly postulated physical entities appear.

axioms (2)
  • standard math Standard axioms of category theory (objects, morphisms, composition, identities)
    Any cobordism category is built inside the framework of categories.
  • domain assumption A variation of stratified Morse theory applies to nested manifolds and produces generators
    Explicitly invoked in the abstract as the source of generators for the general nested cobordism category.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors

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    Introduces uncoiled affine and periodic Temperley-Lieb algebras as finite quotients and constructs explicit Wenzl-Jones idempotents projecting onto their one-dimensional modules, with Markov trace evaluations expresse...

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