Published in Theory and Applications of Categories, 2024
We introduce a string diagram calculus for strict $4$-categories and use it to prove that given a cofinite inclusion of $4$-categorical presentations, the induced restriction functor on mapping spaces to a fixed target strict $4$-category is a fibration of strict $4$-groupoids.
journal (Open access)
Published in cahiers de topologie et géométrie différentielle catégoriques, 2024
An $n$-sesquicategory is an $n$-globular set with strictly associative and unital composition and whiskering operations, which are however not required to satisfy the Godement interchange laws which hold in $n$-categories. In a previous paper we showed how these can be defined as algebras over a monad whose operations are simple string diagrams. In this paper, we give an explicit description of computads for this monad and we prove that the associated category of computads is a presheaf category. We use this to describe a string diagram notation for representing arbitrary composites in $n$-sesquicategories. This is a step towards a theory of string diagrams for semistrict $n$-categories.
journal (Open access)
Published in Compositionality, 2022
We define a coherent adjunction in a strict $3$-category and we use string diagrams to show that any adjunction can be extended to a coherent adjunction in an essentially unique way.
journal (Open access)
Published in Theory and Applications of Categories, 2022
We define a monad on $n$-globular sets whose operations are encoded by simple string diagrams and we define $n$-sesquicategories as algebras over this monad. This monad encodes the compositional structure of $n$-dimensional string diagrams. We give a generators and relations description of this monad, which allows us to describe $n$-sesquicategories as globular sets equipped with associative and unital composition and whiskering operations. One can also see them as strict $n$-categories without interchange laws. Finally we give an inductive characterization of $n$-sesquicategories.
journal (Open access)
with Gustavo Granja
Published in Journal of Homotopy and Related Structures, 2017
Let $X$ be a union of a sequence of symplectic manifolds of increasing dimension and let $M$ be a manifold with a closed $2$-form $\omega$. We use Tischler’s elementary method for constructing symplectic embeddings in complex projective space to show that the map from the space of embeddings of $M$ in $X$ to the cohomology class of ω given by pulling back the limiting symplectic form on $X$ is a weak Serre fibration. Using the same technique we prove that, if $b_2(M)<\infty$, any compact family of closed $2$-forms on $M$ can be obtained by restricting a standard family of forms on a product of complex projective spaces along a family of embeddings.
preprint, 2022
We define a coherent adjunction in a strict $4$-category and we use string diagrams to show that any adjunction can be extended to a coherent adjunction in an essentially unique way.
PhD thesis, Univeristy of Oxford, 2017
supervised by Christopher Douglas
A fully extended framed topological field theory with target in a symmetric monoidal $n$-catgeory $\mathcal{C}$ is a symmetric monoidal functor $Z : \operatorname{Bord}_n \to \mathcal{C}$, where $\operatorname{Bord}_n$ is the symmetric monoidal $n$-category of $n$-framed bordisms. The cobordism hypothesis says that such field theories are classified by fully dualizable objects in $\mathcal{C}$. Given a fully dualizable object $X$ in $\mathcal{C}$, we are interested in computing the values of the corresponding field theory on specific framed bordisms. This leads to the question of finding a presentation for $\operatorname{Bord}_n$. In view of the cobordism hypothesis, this can be rephrased in terms of finding coherence data for fully dualizable objects in a symmetric monoidal $n$-category. We prove a characterization of full dualizability of an object $X$ in terms of existence of a dual of $X$ and existence of adjoints for a finite number of higher morphisms. This reduces the problem of finding coherence data for fully dualizable objects to that of finding coherence data for duals and adjoints. For $n=3$, and in the setting of strict symmetric monoidal $3$-categories, we find this coherence data, and we prove the corresponding coherence theorems. The proofs rely on extensive use of a graphical calculus for strict monoidal $3$-categories.