We introduce a theory of twisted simplicial distributions on simplicial principal bundles, which allow us to capture Bell’s non-locality, and the more general notion of quantum contextuality. We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions in terms of both direct constructions in simplicial sets and techniques from homological algebra. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.

Given an operad O, we define a notion of weak O-monoids – which we term O-pseudomonoids – in a 2-category. In the special case with the 2-category in question is the 2-category 𝖢𝖺𝗍 of categories, this yields a notion of O-monoidal category, which in the case of the associative and commutative operads retrieves unbiased notions of monoidal and symmetric monoidal categories, respectively. We carefully unpack the definition of O-monoids in the 2-categories of discrete fibrations and of category-indexed sets. Using the classical Grothendieck construction, we thereby obtain an O-monoidal Grothendieck construction relating lax O-monoidal functors into Set to strict O-monoidal functors which are also discrete fibrations.

In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of O-monoidal categories, we state and prove a Grothendieck construction for lax O-monoidal functors into convex sets. We apply this construction to the categorical characterization of entropy of Baez, Fritz, and Leinster, and to the study of quantum contextuality in the framework of simplicial distributions.

2023

ArXiv preprint

Frobenius and commutative pseudomonoids in the bicategory of spans

Ivan Contreras, Rajan Amit Mehta, and Walker H. Stern

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in Span correspond, respectively, to paracyclic sets and Γ-sets satisfying the 2-Segal conditions. These results connect closely with work of the third author on A_∞algebras in ∞-categories of spans, as well as the growing body of work on higher Segal objects. Because our motivation comes from symplectic geometry and topological field theory, we emphasize the direct and computational nature of the classifications and their proofs.

In this work, we conclude our study of fibred ∞-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the ∞-bicategory of 2-Cartesian fibrations over S and the ∞-bicategory of contravariant functors \(S^{\text o\text p }\to \mathbb B\text i\text c\text a\text t_∞\) with values in the ∞-bicategory of ∞-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

ACS

2-Cartesian Fibrations I: A Model for ∞-Bicategories Fibred in ∞-Bicategories

In this paper, we provide a notion of ∞-bicategories fibred in ∞-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling ∞-bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an ∞-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of \(\operatornameSet_∆\)-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.

In this work, we prove a generalization of Quillen’s Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on (∞,1)-localizations. When restricted to 1-categories with all morphisms marked, our theorem retrieves the classical Theorem A of Quillen. We additionally state and provide evidence for a new conjecture: the cofinality conjecture, which describes the relation between a conjectural theory of marked (∞,2)-colimits and our generalization of Theorem A.

Top. App.

Topological field theories on open-closed r-spin surfaces

In this article, we establish a connection between two models for r-spin structures on surfaces: the marked PLCW decompositions of Novak and Runkel-Szegedy, and the structured graphs of Dyckerhoff-Kapranov. We use these models to describe r-spin structures on open-closed bordisms, leading to a generators-and-relations characterization of the 2-dimensional open-closed r-spin bordism category. This results in a classification of 2-dimensional open closed field theories in terms of algebraic structures we term “knowledgeable \Lambda_r-Frobenius algebras”. We additionally extend the state sum construction of closed r-spin TFTs from a \Lambda_r-Frobenius algebra A with invertible window element of Novak and Runkel-Szegedy to the open-closed case. The corresponding knowledgeable \Lambda_r-Frobenius algebra is A together with the Z/r-graded center of A.

2021

JHRS

2-Segal objects and algebras in spans

Walker H. Stern

Journal of Homotopy and Related Structures, Jun 2021

We define a category parameterizing Calabi–Yau algebra objects in an infinity category of spans. Using this category, we prove that there are equivalences of infinity categories relating, firstly: 2-Segal simplicial objects in C to algebra objects in \(\operatoname{Span}(C)\); and secondly: 2-Segal cyclic objects in C to Calabi–Yau algebra objects in \(\operatoname{Span}(C)\).

2020

AGT

A relative 2–nerve

Fernando Abellán García, Tobias Dyckerhoff, and Walker H. Stern

In this thesis we analyze 2-dimensional open topological field theories in both 1-categorical and ∞-categorical contexts. Making use of the formalism, introduced by Dyckerhoff and Kapranov, of graphs structured over a crossed simplicial group \(∆\mathfrak G\), we give combinatorial models for 2-dimensional open cobordism categories with additional structure — orientations, N-spin structures, etc. We then use this model to effect a classification of the corresponding classes of 1-categorical topological field theories. This classification retrieves, in special cases, a number of results known in the literature, as well as providing new results. We then turn to 2-dimensional open oriented topological field theories valued in an ∞-category \(\operatorname{Span}(C)\)of spans in an ∞-category C. Applying a theorem stated by Lurie in [33], such topological field theories are classified by Calabi-Yau algebras in \(\operatorname{Span}(C)\). We define two 1-categories whose functors to C parameterize, respectively, associative algebras and Calabi-Yau algebras in \(\operatorname{Span}(C)\). We prove that there is an equivalence of ∞-categories between associative algebras in Span(C) and 2-Segal simplicial objects in C; and we prove an equivalence of ∞-categories between Calabi-Yau algebras in \(\operatorname{Span}(C)\)and 2-Segal cyclic objects in C. We discuss the invariants the resultant topological field theories assign to surfaces, and develop the example provided by cyclic structures on Cech nerves.

2016

arXiv

Structured Topological Field Theories via Crossed Simplicial Groups