Coherence for adjunctions in a 3-category via string diagrams
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)
Simple string diagrams and n-sesquicategories
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)
Symplectic embeddings in infinite codimension
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 ω. 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 b2(M)<∞, 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.
Computads and string diagrams for n-sesquicategories
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.
Coherence for adjunctions in a 4-category
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.
String diagrams for 4-categories and fibrations of mapping 4-groupoids
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.
Coherence for 3-dualizable objects
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 C is a symmetric monoidal functor Z from Bord(n) to C, where 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 C. Given a fully dualizable object X in 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 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.
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