David Reutter

With Giovanni Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Lukas Müller, Nivedita, David Penneys, Claudia Scheimbauer, Luuk Stehouwer and Chetan Vuppulury

We present a coherent definition of dagger (∞,n)-category in terms of equivariance data trivialized on parts of the category. Our main example is the bordism higher category Bord_n^X. This allows us to define a reflection-positive topological quantum field theory to be a higher dagger functor from Bord_n^X to some target higher dagger category C. Our definitions have a tunable parameter: a group G acting on the  (∞,1)-category Cat_{(∞,n)} of (∞,n)-categories. Different choices for G accommodate different flavours of higher dagger structure; the universal choice is G = Aut(Cat_{(∞,n)}) =(ℤ/2ℤ)^n, which implements dagger involutions on all levels of morphisms. 

The Stratified Cobordism Hypothesis suggests that there should be a map PL(n)→Aut(AdjCat_{(∞,n)}), where PL(n) is the group of piecewise-linear automorphisms of R^n and AdjCat_{(∞,n)} the  (∞,1)-category of (∞,n)-categories with all adjoints; we conjecture more strongly that Aut(AdjCat_{ (∞,n)}) ≅ PL(n). Based on this conjecture we propose a notion of dagger  (∞,n)-category with unitary duality or PL(n)-dagger category. We outline how to construct a PL(n)-dagger structure on the fully-extended bordism  (∞,n)-category Bord_n^X for any stable tangential structure X; our outline restricts to a rigorous construction of a coherent dagger structure on the unextended bordism  (∞,1)-category Bord_{n,n-1}^X.  

The article is a report on the results of a workshop held in Summer 2023, and is intended as a sketch of the big picture and an invitation for more thorough development.

arXiv 2024

With Yu Leon Liu, Aaron Mazel-Gee, Catharina Stroppel and Paul Wedrich

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and 4. Here we provide the first categorification of this Hecke braided monoidal category, which takes the form of an 𝔼2-monoidal (∞,2)-category whose hom-(∞,1)-categories are k-linear, stable, idempotent-complete, and equipped with ℤ-actions. This categorification is designed to control homotopy-coherent link homology theories and to-be-constructed topological quantum field theories in dimension 4 and 5. 

Our construction is based on chain complexes of Soergel bimodules, with monoidal structure given by parabolic induction and braiding implemented by Rouquier complexes, all modelled homotopy-coherently. This is part of a framework which allows to transfer the toolkit of the categorification literature into the realm of ∞-categories and higher algebra. Along the way, we develop families of factorization systems for (∞,n)-categories, enriched ∞-categories, and ∞-operads, which may be of independent interest. 

As a service aimed at readers less familiar with homotopy-coherent mathematics, we include a brief introduction to the necessary ∞-categorical technology in the form of an appendix.

arXiv 2024

With Christopher Schommer-Pries

Extending the work of the first author, we introduce a notion of semisimple topological field theory in arbitrary even dimension and show that such field theories necessarily lead to stable diffeomorphism invariants. The main result of this paper is a proof that this 'upper bound' is optimal: To this end, we introduce and study a class of semisimple topological field theories, generalizing the well known finite gauge theories constructed by Dijkgraaf-Witten, Freed and Quinn. We show that manifolds satisfying a certain finiteness condition -- including 4-manifolds with finite fundamental group -- are indistinguishable to these field theories if and only if they are stably diffeomorphic. Hence, such generalized Dijkgraaf-Witten theories provide the strongest semisimple TFT invariants possible. These results hold for a large class of ambient tangential structures.

We discuss a number of applications, including the constructions of unoriented 4-dimensional semisimple field theories which can distinguish unoriented smooth structure and oriented higher-dimensional semisimple field theories which can distinguish certain exotic spheres. 

Along the way, we show that dimensional reductions of generalized Dijkgraaf-Witten theories are again generalized Dijkgraaf-Witten theories, we utilize ambidexterity in the rational setting, and we develop techniques related to the ∞-categorical Möbius inversion principle of Gálvez-Carrillo--Kock--Tonks.

arXiv 2022

With Christopher Dean, Eric Finster, Ioannis Markakis and Jamie Vicary.

We give a new description of computads for weak globular ω-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads, and allows a new definition of ω-category that avoids the technology of globular operads. Our framework permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with variable-to-variable maps is a presheaf topos, giving a direct description of the index category. We prove that our resulting definition of ω-category agrees with that of Batanin and Leinster and that the induced notion of cofibrant replacement for ω-categories coincides with that of Garner.

arXiv 2022

Advances in Mathematics 450:109739, 2024

With Lukas Heidemann and Jamie Vicary

The theory of associative n-categories has recently been proposed as a strictly associative and unital approach to higher category theory. As a foundation for a proof assistant, this is potentially attractive, since it has the potential to allow simple formal proofs of complex high-dimensional algebraic phenomena. However, the theory relies on an implicit term normalisation procedure to recognize correct composites, with no recursive method available for computing it. 

Here we describe a new approach to term normalisation in associative n-categories, based on the categorical zigzag construction. This radically simplifies the theory, and yields a recursive algorithm for normalisation, which we prove is correct. Our use of categorical lifting properties allows us to give efficient proofs of our results. This normalisation algorithm forms a core component of the proof assistant homotopy.io, and we illustrate our scheme with worked examples.

arXiv 2022

Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022)

With Theo Johnson-Freyd

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension. As a corollary, every pseudounitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. 

Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in H^5 (K (Z2 , 2); k^x) and we use a numerical invariant -- defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -- to prove that this obstruction always vanishes. 

Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the known model of Z(Mod-B) in terms of braided B-module categories, we introduce a new computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field.

arXiv 2021

Journal of the American Mathematical Society 37:81-150, 2024

With Corey Jones and David Penneys

A braided monoidal category may be considered a 3-category with one object and one 1-morphism. In this paper, we show that, more generally, 3-categories with one object and 1-morphisms given by elements of a group G correspond to G-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of 3-categories C equipped with a 3-functor BG→C which is essentially surjective on objects and 1-morphisms is equivalent to the 2-category of G-crossed braided categories. This provides a uniform approach to various constructions of G-crossed braided categories.

arXiv 2020

Journal of the London Mathematical Society 107(1):333-406, 2023

With Eric Finster and Jamie Vicary

We present a type theory for strictly unital ∞-categories, in which a term computes to its strictly unital normal form. Using this as a toy model, we argue that it illustrates important unresolved questions in the foundations of type theory, which we explore. Furthermore, our type theory leads to a new definition of strictly unital ∞-category, which we claim is stronger than any previously described in the literature.

arXiv 2020

Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, 2022 (LICS 2022) 

We prove that semisimple 4-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth 4-manifolds and homotopy equivalent simply connected closed oriented smooth 4-manifolds. We show that all currently known 4-dimensional field theories are semisimple, including unitary field theories, and once-extended field theories which assign algebras or linear categories to 2-manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4-manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple 4-dimensional field theory is invariant under ℂP^2-stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the `point particles' of the field theory. Such fermion-free field theories cannot distinguish homotopy equivalent 4-manifolds. Throughout, we illustrate our results with the Crane-Yetter-Kauffman field theory associated to a ribbon fusion category. As an algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.

arXiv 2020

Journal of Topology 16(2):542-566, 2023

A unitary fusion category is a fusion category over the complex numbers with a compatible positive dagger structure. Such unitary structures naturally arise in many applications and constructions of fusion categories, most notably in the context of operator algebras, subfactor theory, and mathematical physics. Not every fusion category admits a unitary structure, and a fusion category could, in principle, have more than one unitary structure. Especially in light of the recent powerful operator algebraic classification techniques of unitary fusion categories, the question of uniqueness of unitary structure has become a significant open problem. In this paper, we prove that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories.

arXiv 2019

Communcations in Mathematical Physics 397:37-52, 2023

With Jamie Vicary. 

The theory of associative n-categories (ANC) is a model for semistrict higher categories, recently developed by Christoph Dorn, Christopher Douglas, and Jamie Vicary, and based on a combinatorial representation of higher string diagrams. In this work, we continue the development of ANCs and introduce the proof assistant homotopy.io. We develop a general contraction principle in ANCs which forms the mathematical heart of homotopy.io, allowing string diagram manipulations in arbitrary dimensions.

arXiv 2019

Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, 2019 (LICS 2019) 

With Christopher Douglas.

One of the early successes of quantum topology was the construction of a three-dimensional topological field theory by Turaev-Viro, and its generalization by Barrett-Westbury, starting from the data of a spherical fusion category. In this work, we introduce spherical fusion 2-categories and construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds, which categorifies the Turaev-Viro-Barrett-Westbury invariant. There are many examples of fusion 2-categories, including the 2-category of module categories of a braided fusion category, the 2-category of 2-representations of a finite 2-group and twisted linearizations of finite 2-groups. 

arXiv 2018

To appear in Memoirs of the AMS.

With Benjamin Musto and Dominic Verdon.

Using the machinery developed in `A compositional approach to quantum functions', we classify instances of quantum pseudo-telepathy in the graph isomorphism game by showing that graphs quantum isomorphic to a graph G correspond to certain Frobenius algebras in a monoidal category of quantum graph automorphisms of G. In certain cases, this classification can be expressed in group-theoretical terms.

arXiv 2018

Communications in Mathematical Physics, 365(2):797-845, 2019

With Benjamin Musto and Dominic Verdon.

Pseudo-telepathy is a phenomenon in quantum information, where two non-communicating parties can use pre-shared entanglement to perform a task classically impossible without communication. In this work, we uncover a connection between pseudo-telepathy and quantum symmetry, expressed in the language of fusion categories. We use this connection to classify instances of pseudo-telepathy in the `graph isomorphism game'. In this case, the relevant categories are (co)representation categories of `quantum automorphisms groups' of graphs, studied in compact quantum group theory. Along the way, we develop a 2-categorical framework for finite quantum set theory.

arXiv 2018

Journal of Mathematical Physics, 59(8):081706, 42, 2018

With Jamie Vicary.

We give a tangle-based graphical language for the description of quantum circuits such that isotopic tangles yield equivalent programs. This leads to several new protocols and topological insight into known phenomena such as error correcting codes and local inversion of cluster states. We analyze 11 known quantum procedures in this way -- including entanglement manipulation, error correction and teleportation -- and in each case present a fully-topological formal verification, yielding generalized procedures in some cases. 

arxiv 2018

Proceedings of the Royal Society A, 475(2224):20180338, 2019

With Jamie Vicary.

We describe a certain combinatorial 2-category and use it to give a mathematical analysis of a new type of classical computer network architecture, intended as a model of a new technology that has recently been proposed in industry. Our approach is based on groubits, generalizations of classical bits based on groupoids. This network architecture allows the direct execution of a number of protocols that are usually associated with quantum networks, including teleportation, dense coding and secure key distribution.

arXiv 2017

Logical Methods in Computer Science, 15(1), 2019 

With Jamie Vicary.

Biunitary connections are central tools in the study and classification of subfactors. At the same time, several important quantum informatic quantities such as Hadamard matrices, unitary error bases and quantum Latin squares can be expressed as special cases of biunitaries. This leads to a range of applications of biunitaries in various 2-categories to quantum and classical information theory. In this paper, we use the interpretation of Hadamard matrices, unitary error bases and quantum Latin squares as biunitaries to obtain many new construction methods for these quantities. An extended abstract can be found here.

arXiv 2016

Higher Structures, 3(1):109–154, 2019