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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

These are lecture notes I wrote for talks by Kevin Costello at the Winter School in Mathematical Physics 2012.