We continue our study of semi-strict tricategories in which the only weakness is in vertical composition. We assemble the doubly-degenerate such tricategories into a 2-category, defining weak functors and transformations. We exhibit a biadjoint biequivalence between this 2-category and the 2-category of braided (weakly) monoidal categories, braided (weakly) monoidal functors, and monoidal transformations.
Alex Corner
Category Theory
My current interests in category theory lie primarily in the world of higher categories, particularly in questions of coherence and generalisations of classical ideas to these higher settings.
My current focus is a project together with Eugenia Cheng to investigate weak vertical composition in the context of semi-strict tricategories. The preprints below discuss this in the case of doubly-degenerate tricategories where we have extended results to an equivalence of totalities, as a stepping stone on the way to investigating doubly-degenerate Trimble 3-categories.
I am also involved in a project to tie together the preprint with Nick Gurski below and a further preprint of Gurski's, based around the central theme of action operads.
Further to this I am working on refining some results from my PhD thesis, thinking about semi-strict monoidal bicategories rather than fully weak monoidal bicategories directly. An area I would like to understand more is categorical generalisations of discrete finite automata.
Publications
We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category \(B\) there is a doubly-degenerate vertically weak semi-strict tricategory whose associated braided monoidal category is braided monoidal equivalent to \(B\).
In this work we define a 2-dimensional analogue of extranatural transformation and use these to characterise codescent objects. They will be seen as universal objects amongst pseudo-extranatural transformations in a similar manner in which coends are universal objects amongst extranatural transformations. Some composition lemmas concerning these transformations are introduced and a Fubini theorem for codescent objects is proven using the universal characterisation description.
Preprints
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in \(\mathbf{Cat}\). We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.
PhD Thesis
Ends and coends can be described as objects which are universal amongst extranatural transformations. We describe a categorification of this idea, extrapseudonatural transformations, in such a way that bicodescent objects are the objects which are universal amongst such transformations. We recast familiar results about coends in this new setting, providing analogous results for bicodescent objects. In particular we prove a Fubini theorem for bicodescent objects. The free cocompletion of a category \(\mathcal{C}\) is given by its category of presheaves \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\). If \(\mathcal{C}\) is also monoidal then its category of presheaves can be provided with a monoidal structure via the convolution product of Day. This monoidal structure describes \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) as the free monoidal cocompletion of \(\mathcal{C}\). Day’s more general statement, in the \(\mathcal{V}\)-enriched setting, is that if \(\mathcal{C}\) is a promonoidal \(\mathcal{V}\)-category then \([\mathcal{C}^{\mathrm{op}}, \mathcal{V}]\) possesses a monoidal structure via the convolution product. We define promonoidal bicategories and go on to show that if \(\mathcal{A}\) is a promonoidal bicategory then the bicategory of pseudofunctors \(\mathbf{Bicat}(\mathcal{A}^{op} ,\mathbf{Cat})\) is a monoidal bicategory.
Extras
This plenary talk was presented at the International Category Theory Conference at Universidade de Santiago de Compostela.
This poster was presented at the Algebraic Topology Seminar at the University of Sheffield.
This poster was presented at the International Category Theory Conference at UCLouvain in Louvain-la-Neuve.
This blog post was written as part of the first Kan Extension Seminar organised by Emily Riehl.
Mathematics Education
Along with Peter Rowlett, I am collecting data for a project concerned with students' perception of their own identities as mathematicians/learners of mathematics.
I am also interested in students' understanding of the role of assumptions in pure mathematics and how this can be compared to their understanding in a more applied setting, such as that of mathematical modelling.
Publications
Due to the nature of the teaching environment, students may often develop perceptions of their lecturers’ ability as mathematicians, based on the pre-prepared and well-rehearsed content they present. In reality, performing mathematical calculations and solving problems is a difficult skill, and students may compare their own experiences unfavourably with the ease they see lecturers display. To interrogate this disparity, an exercise was included in an undergraduate maths session during which lecturers attempted unseen problems from A-level maths papers, so the students could see them model the process of problemsolving - including making istakes, applying helpful strategies and techniques, correcting their own errors, and identifying gaps in their knowledge. As well as modelling these positive behaviours for students, the session aimed to develop the students’ understanding that their own experiences of struggling with maths are normal and healthy. The activity formed part of a broader session on making mistakes in maths, which also included some advice and opportunities to find mistakes in mathematical working-out. The students were invited to participate in questionnaires and focus groups to explore their perceptions and attitudes towards their lecturers’ knowledge and capacity to make mistakes, and in this paper we analyse these responses and consider how this relates to teaching, and to students’ personal development.
During the COVID-19 pandemic, the teaching of programming for undergraduate mathematicians was moved online. This was delivered asynchronously, with students working through notes and exercises and asking for help from staff via online messages as needed. Staff delivery time was redirected from content delivery into a formal system of formative assessment, which replaced informal discussion and feedback during in-person classes. Formative tasks were submitted and feedback was provided via GitHub Classroom. Students were broadly positive about the formative feedback system and mixed about the need for live delivery. Formal formative feedback highlighted that students may hold incorrect views about the accuracy of task completion, making formal formative submission an effective use of staff delivery time.
Topic modelling, an automated literature review technique, is used to generate a list of topics from the text of all articles published in previous issues of MSOR Connections. There are many topics of consistent popularity, including assessment, employability, school-university transition and the teaching of specific subjects and skills with the mathematics, statistics and operational research disciplines. We identify some topics that have waned in popularity, especially following the demise of the MSOR Network, including organised book and software reviews, conference and workshop announcements and reports, and articles focused on staff development. In its present form as a fully peer-reviewed practitioner journal, there appears to be a shift in focus from personal reflection to evidence-based research. There is a high focus on innovative practice using technology in the publication, though with less focus on specific software over time. Similarly, more nuance appears to be entering the discourse over maths support and e-assessment as these topics mature. We note a rise over time in student-centred approaches and a sudden rise in the previous issue of digital and remote learning due to the COVID-19 pandemic. We speculate about future trends that may emerge, including an increased focus on digital and remote learning and an increase in content on equity, equality, diversity and inclusion.
A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.
Problems based on applications or objects were added into a first year pure module in gaps where real-life problems were missing. Physical props were incorporated within the teaching sessions where it was possible. The additions to the module were the utilities problem whilst studying planar graphs, data storage when looking at number bases, RSA encryption after modular arithmetic and the Euclidean algorithm, as well as molecules and the mattress problem when looking at group theory. The physical objects used were tori, molecule models and mini mattresses. Evaluation was carried out through a questionnaire to gain the students' opinions of these additions and their general views of applications. Particular attention was paid to the effect on engagement and understanding.
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