David Yetter - Academia.edu (original) (raw)

Papers by David Yetter

arXiv (Cornell University), Feb 25, 2016

We derive the general state sum construction for 2D topological quantum field theories (TQFTs) wi... more We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda & Pfeiffer [11]). From the extended Pachner moves (Crane & Yetter [4]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [5], we require that triangulations be flag-like, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flag-likeness.

arXiv (Cornell University), Jul 20, 2018

We apply the theory of directed topology developed by Grandis [9, 10] to the study of stratified ... more We apply the theory of directed topology developed by Grandis [9, 10] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [5, 8], which we extend to a similar construction of a Dijkgraaf-Witten type TQFT in the case where the defects (lower dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orientation convention.

arXiv (Cornell University), Sep 1, 1998

arXiv (Cornell University), Mar 20, 2023

arXiv (Cornell University), Oct 24, 2014

The k-graphs in the sense of Kumjian and Pask [7] are discrete Conduché fibrations over the monoi... more The k-graphs in the sense of Kumjian and Pask [7] are discrete Conduché fibrations over the monoid N k in which ever morphism in the base has a finite preimage under the fibration. We examine the generalization of this construction to discrete Conduché fibrations with the same finiteness condition and a lifting property for completions of cospans to commutative squares, over any category satisfying a strong version of the right Ore condition, including all categories with pullbacks and right Ore categories in which all morphisms are monic.

Involve, Mar 4, 2021

We consider the problem of when one quandle homomorphism will factor through another, restricting... more We consider the problem of when one quandle homomorphism will factor through another, restricting our attention to the case where all quandles involved are connected. We provide a complete solution to the problem for surjective quandle homomorphisms using the structure theorem for connected quandles of Ehrman et al. [2] and the factorization system for surjective quandle homomorphsims of Bunch et al. [1] as our primary tools. The paper contains the substantive results obtained by an REU research group consisting of the first four authors under the mentorship of the fifth, and was

Journal of Knot Theory and Its Ramifications, Apr 1, 2017

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot int... more We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf-Witten theory. The appropriate initial data for the construction are certain three object categories with coefficients satisfying a partially degenerate cocycle condition.

arXiv (Cornell University), May 12, 2003

Using the theory of measurable categories developped in [Yet03], we provide a notion of represent... more Using the theory of measurable categories developped in [Yet03], we provide a notion of representations of 2-groups more well-suited to physically and geometrically interesting examples than that using 2-VECT (cf. [KV94]). Using this theory we sketch a 2-categorical approach to the state-sum model for Lorentzian quantum gravity proposed in [CY03], and suggest state-integral constructions for 4-manifold invariants.

arXiv (Cornell University), May 14, 2002

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometr... more We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of homomorphisms from a(n augmented) knot quandle associated with the base to a suitable (augmented) quandle associated with the fiber. We consider the cases of the monodromy of a branched covering, braid monodromy and the monodromy of a Lefschetz fibration. The present paper is an expanded and corrected version of [Yet02].

arXiv (Cornell University), Sep 16, 2018

The theory of welded and extended welded knots is a generalization of classical knot theory. Weld... more The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeistertype moves. We present a new class of invariants for welded and extended welded knots and links using a multi-skein relation, following Z. Yang's approach for virtual knots. Using this skein-theoretic approach, we find sufficient conditions on the coefficients to obtain invariance under the extended Reidemeister moves appropriate to welded and extended welded links.

arXiv (Cornell University), Dec 2, 1994

arXiv (Cornell University), Sep 27, 1994

We provide, with proofs, a complete description of the authors' construction of state-sum invaria... more We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of Rep(U q (sl 2)) (q a principal 4r th root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel.

arXiv (Cornell University), Jan 28, 1998

In [1] Barrett and Crane introduce a modification of the generalized Crane-Yetter state-sum (cf. ... more In [1] Barrett and Crane introduce a modification of the generalized Crane-Yetter state-sum (cf. [2]) based on the category of representations of Spin(4) ∼ = SU (2) × SU (2), which provides a four-dimensional analogue of Regge and Ponzano's [9] spin-network formulation of three-dimensional gravity. The key to the modification of the Crane-Yetter state-sum is the use of a different intertwiner between the "inbound" and "outbound" tensor products of objects assigned to faces of the tetrahedron, thereby imposing a "quantum analogue" of the condition that the sum of the simple bivectors represented by two faces with a common edge is itself simple. The purposes of this paper are

arXiv (Cornell University), Dec 6, 2005

This paper summarizes substantive new results derived by a student team (the first three authors)... more This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU ``Brainstorming and Barnstorming''. The main results are a decomposition theorem for quandles in terms of an operation of `semidisjoint union' showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribe inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

arXiv (Cornell University), Sep 24, 2007

We adapt the work of Power [14] to describe general, not-necessarily composable, not-necessarily ... more We adapt the work of Power [14] to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in the 2-category of k-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack [9], proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov [10]. The author wishes to thank Kansas State University and the University of Pennsylvania for support of the sabbatical leave when this work was begun. The author also wishes to than the referee assigned by Theory and Applications of Categories for numerous corrections and helpful suggestions for improvements to the original manuscript.

Cornell University - arXiv, Jun 28, 2019

We solve a very classical problem: providing a description of the geometry of a Euclidean tetrahe... more We solve a very classical problem: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles and (an) edge length, the remaining parts being derivable by symmetry or by identities in the classic 1902 compendium of results on tetrahedral geometry by G. Richardson. We also provide an alternative proof using (bi)vectors of the result of Yetter that four times the sum of the squared areas of the medial parallelograms is equal to the sum of the squared areas of the faces. Despite the classical nature of the problem, it would not have been natural to consider had it not been suggested by recent work in quantum physics.

Journal of Knot Theory and Its Ramifications

We apply the theory of directed topology developed by Grandis [Directed homotopy theory, I. The f... more We apply the theory of directed topology developed by Grandis [Directed homotopy theory, I. The fundamental category, Cah. Topol. Géom. Différ. Catég. 44 (2003) 281–316; Directed Algebraic Topology[Formula: see text] Models of Non-Reversible Worlds, New Mathematical Monographs, Vol. 13 (Cambridge University Press, Cambridge, 2009)] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [A. L. Dougherty, H. Park and D. N. Yetter, On 2-dimensional Dikjgraaf-Witten theory with defects, J. Knot Theory Ramifications 25(5) (2016) 1650021, doi:10.1142/S0218216516500218; I. J. Lee and D. N. Yetter, Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds, J. Knot Theory Ramifications 26(5) (2017) 1750026, doi:10.1142/S0218216517500262], which we ex...

$Research paper thumbnail of Weakly <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>ω-Categorified Models of Algebraic Theories$

arXiv: Category Theory, 2020

We provide the expected constructions of weakly omega\omegaomega-categorified models (in the sense of Bre... more We provide the expected constructions of weakly omega\omegaomega-categorified models (in the sense of Bressie) of the theory of groups and quandles which arise by replacing the homotopies used to give equivalence relations in the theory of fundamental groups, fundamental quandles, and knot quandles with homotopies of all orders used as arrows of categorical dimensions one and greater, and discuss other related constructions of weakly omega\omegaomega-categorifed algebras.