Claudia I. Scheimbauer
adjunction adjunction
oplax oplax

research interests

My research area is mathematical physics. I work on fully extended topological field theories (in the sense of Lurie) using higher category theory, factorization algebras/homology, and derived symplectic geometry. I am also interested in relative, or twisted, field theories in the sense of Freed-Teleman and Stolz-Teichner, in particular as realizing (possibly non-invertible) symmetries. I have worked on several questions revolving around objects carrying a higher categorical structure called "2-Segal object" which was introduced by Dyckerhoff-Kapranov and Gàlvez-Carrillo-Kock-Tonks and generalizes the idea of a categorical structure with a multivalued composition. These questions include applications in topology (Waldhausen construction and K-theory, configuration spaces) and mathematical physics (modular functors).


Assembly of Constructible Factorization Algebras, arXiv:2403.19472
with Eilind Karlsson and Tashi Walde

    We provide a toolbox of extension, gluing, and assembly techniques for factorization algebras. Using these tools, we fill various gaps in the literature on factorization algebras on stratified manifolds, the main one being that constructible factorization algebras form a sheaf of symmetric monoidal ∞-categories. Additionally, we explain how to assemble constructible factorization algebras from the data on the individual strata together with module structures associated to the relative links; thus answering a question by Ayala. Along the way, we give detailed proofs of the following facts which are also of independent interest: constructibility is a local condition; the ∞-category of disks is a localization of any sufficiently fine poset of disks; constructibility implies the Weiss condition on disks. For each of these, variants or special cases already existed, but they were either incomplete or not general enough.

Dagger n-categories, arXiv:2403.01651
with Giovanni Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Lukas Müller, Nivedita, David Penneys, David Reutter, Luuk Stehouwer, 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 BordXn. This allows us to define a reflection-positive topological quantum field theory to be a higher dagger functor from BordXn 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 Rn 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 BordXn for any stable tangential structure X; our outline restricts to a rigorous construction of a coherent dagger structure on the unextended bordism (∞,1)-category BordXn,n−1. 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.

Relative field theories via relative dualizability, arxiv:2312.0505
with Thomas Stempfhuber

    We investigate relative versions of dualizability designed for relative versions of topological field theories (TFTs), also called twisted TFTs, or quiche TFTs in the context of symmetries. In even dimensions we show an equivalence between lax and oplax fully extended framed relative topological field theories valued in an (∞,N)-category in terms of adjunctibility. Motivated by this, we systematically investigate higher adjunctibility conditions and their implications for relative TFTs. Summarizing we arrive at the conclusion that oplax relative TFTs is the notion of choice. Finally, for fun we explore a tree version of adjunctibility and compute the number of equivalence classes thereof.

The AKSZ construction in derived algebraic geometry as an extended topological quantum field theory, accepted for publication in Memoirs of the AMS, arxiv:2108.02473
with Damien Calaque and Rune Haugseng

    We combine tools from extended topological field theories, derived algebraic geometry and higher categories to prove that the AKSZ construction of shifted symplectic forms on σ-models, i.e.~derived mapping stacks extends to a fully extended semi-classical TFT. This extends work by Pantev—Toën—Vaquié—Vezzosi and Calaque. The construction is two fold: on the one hand, they show full dualizability and use the Cobordism Hypothesis; on the other hand, they exhibit the symmetric monoidal functor in question, thus demonstrating the Cobordism Hypothesis in action. Examples include classical Chern–Simons theory, classical Rozansky–Witten theory, and the Poisson sigma model.

Comparison of Waldhausen constructions, Ann. K-Theory 6 (2021), no. 1, 97–136, arxiv:1901.03606
with Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, and Martina Rovelli

    We generalize our previous, discrete, result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an S-construction.

2-Segal objects and the Waldhausen construction, Algebr. Geom. Topol. 21 (2021), no. 3, 1267–1326, arxiv:1809.10924
with Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, and Martina Rovelli

    We generalize our previous, discrete, result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an S-construction.

The edgewise subdivision criterion for 2-Segal objects, Proc. Amer. Math. Soc., 148(1):71–82, arxiv:1807.05069
with Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, and Martina Rovelli

    We show that the edgewise subdivision of a 2-Segal object is always a Segal object, and furthermore that this property characterizes 2-Segal objects.

Duals and adjoints in the factorization higher Morita category arxiv:1804.10924
with Owen Gwilliam

    We study dualizability in the factorization (∞,n+N)-Morita category Alg_n(S) of E_n-algebras, bimodules in E_{n-1}-algebras, bimodules of bimodules, etc. in an (∞,N)-category S. More precisely, we that the symmetric monoidal (∞,n)-category underlying Alg_n(S) is fully n-dualizable, i.e. every object has a dual and every k-morphism has a left and a right adjoint for 0 < k < n. The Cobordism Hypothesis gives an immediate application of this result, namely the existence of categorified n-dimensional topological field theories (which were constructed explicitly in my PhD thesis using factorization homology, see below), and relative versions thereof. The motivation for this result was to construct low-dimensional examples of relative, "twisted", field theories in a subsequent article.

2-Segal sets and the Waldhausen construction, Topology and its Applications 23 (2018) pp. 445-484, arxiv:1609.02853
with Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, and Martina Rovelli

    We give a variant of Waldhausen's S-construction which allows to take in certain double categories and showed that this is an equivalence of categories to the category of multivalued categories (unital 2-Segal sets). An extension of this construction to the homotopical setting should appear soon in a follow-up project.

(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories, Advances in Mathematics 307 (2017) pp. 147-223, arxiv:1502.06526
with Theo Johnson-Freyd

    The main motivation for this project was to give a precise definition of twisted field theories in the setting of higher categories. This led to the development of a framework for lax and oplax transformations and their higher analogs between strong (∞,n)-functors. Another motivation was to give a construction of the higher Morita category of E_d-algebras in a symmetric monoidal (∞,n)-category C as an (∞,n+d)-category, thus extending the construction of my thesis below. Examples include the (∞,4)-category of braided monoidal (nice, linear) categories, monoidal bimodule categories, bimodule categories, functors, and natural transformations.

A note on the (∞,n)-category of cobordisms, Algebr. Geom. Topol., 19(2):533–655, 2019, arxiv:1509.08906
with Damien Calaque

    Inspired by Lurie's paper on the cobordism hypothesis we define an n-fold Segal space of cobordisms, which is a good model for the (∞,n)-category of cobordisms.

Lectures on mathematical aspects of (twisted) supersymmetric gauge theories, Mathematical Aspects of Quantum Field Theories, Damien Calaque and Thomas Strobl, editors, Mathematical Physics Studies. Springer International Publishing, 2015, arxiv:1401.2676
with Kevin Costello

in preparation


Factorization homology as a fully extended topological field theory

We first give a precise definition of a fully extended n-dimensional topological field theory using complete n-fold Segal spaces as a model for (∞,n)-categories and then, given an E_n-algebra A, we explicitly construct a fully extended TFT given by taking factorization homology with coefficients in A. This is the fully extended n-TFT corresponding (via the cobordism hypothesis) to the E_n-algebra A, which is a fully dualizable object in a suitable Morita-(∞,n)-category Alg_n of E_n -algebras.

Some notes from my talk at the Winter School in Mathematical Physics 2014 in Les Diablerets. A video of a talk I gave about my thesis is available here.

Here is the current version of my thesis on "Factorization Homology as a Fully Extended Topological Field Theory".

extended abstracts and slides