Fourth year MMath project on the Dold-Kan correspondence, exploring and explaining the equivalence of categories, and moreover the Quillen Equivalence between simplicial abelian groups and non-negatively graded chain complexes of abelian groups.
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Published in , 2024
We give an elementary description of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ of internal categories, functors and natural transformations, where $\mathcal{E}$ is a category modelling Lawvere’s elementary theory of the category of sets (ETCS). This extends Bourke’s characterisation of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ where $\mathcal{E}$ has pullbacks to take account for the extra properties in ETCS, and Lawvere’s characterisation of the (one dimensional) category of small categories to take account of the two-dimensional structure. Important two-dimensional concepts which we introduce include $2$-well-pointedness, full-subobject classifiers, and the categorified axiom of choice. Along the way, we show how generating families (resp. orthogonal factorisation systems) on $\mathcal{E}$ give rise to generating families (resp. orthogonal factorisation systems) on $\mathbf{Cat}\left(\mathcal{E}\right)_{1}$, results which we believe are of independent interest.
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Slides for a 20 minute talk to first and second year undergraduate students introducing the ideas behind (simplicial) homology and homotopy in a visual way.
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Slides for a 15 minute talk outlining the work I completed on my fourth year MMath project on the Dold-Kan correspondence.
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A 2 hour talk presenting ideas from Richard Garner’s paper “Understanding the Small object Argument”.
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A 1 hour talk presenting ideas from Emily Riehl.
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A talk presenting ideas from Elena Dimitriadis Bermejo’s PhD thesis “A New Model for dg Categories”.
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I explain some of the ideas that were presented at the Homotopy Type Theory conference 2023 at Carnegie Mellon University. In particular, I explain the construction of Riehl and Shulman’s theory of synthetic $(\infty,1)$-categories through their simplicial type theory. I also discuss synthetic algebraic geometry, other extensions of homotopy type theory, and the unifying theory of modal type theory.
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It might not be too much of a stretch to say that we live in a golden age of mathematics; there is an explosion of content, theorems, and disparate research areas. With this vast wealth of content comes the need to check it all. This process is highly fallible and time consuming if done by humans, and so it is natural to ask if there is some way to get a computer to do this for us. However, our set-theoretic foundation presents many barriers to the success of this. So, we rip it up and start again!
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A 30 minute hybrid talk/ mathematical activity in which I explained the maths behind the recent paper by David Cushing and David I. Stewart titled “You need 27 tickets to guarantee a win on the UK National Lottery”
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Abstract: Lawvere’s Elementary Theory of the Category of Sets (ETCS) characterises when a category E behaves ‘similarly’ to the category of sets. More formally, it is an axiomitization of categorical models of ZFC except for the axiom schema of replacement. As such, it provides the language for a category-theoretic foundation of mathematics. In this talk, I will define relevant category-theoretic notions and give an idea of how these relate to the ZFC axioms. I will also exemplify how one can do logic internal to a category satisfying ETCS (or more generally, in an elementary topos). No prior category theoretic knowledge is required. This talk is a precursor to some work that I have been doing with Arian on developing a 2-dimensional version of this; the elementary theory of the 2-category of small categories.
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Quillen model categories provide an abstract framework in which to do homotopy theory. A key example is the folk model structure on $\mathbf{Cat}$, for which the notion of homotopy is that of equivalences of categories. The construction of this model structure requires the axiom of choice, meaning that groupoidal models of type theory which exploit the structure of the trivial cofibrations are not constructive. In analogy to work by Gambino, Henry, Sattler and Szumiło on the effective model structure on simplicial objects in a lextensive category, we construct a model structure on internal categories that does not require the axiom of choice; when we consider categories internal to $\mathbf{Set}$ and assume the axiom of choice, this recovers the folk model structure on $\mathbf{Cat}$. Moreover, we prove that this forms an algebraic model structure whose underlying ordinary model structure turns out to recover Everaert, Kieboom and Van de Linden’s folk model structure on internal categories equipped with the trivial topology. Restricting this construction to a model structure on internal groupoids, the extra algebraicity provides us with a constructive model of Martin-Löf Type Theory due to the type-theoretic algebraic weak factorisation systems of Gambino and Larrea. In this talk, I will introduce the construction of this algebraic model structure and some related ideas.
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Abstract: Lawvere’s Elementary Theory of the Category of Sets (ETCS) provides a set theory which axiomatises the properties of function composition rather than those of a global set membership relation. It provides an important fragment of a category-theoretic foundation of mathematics but is strictly weaker than the traditional foundation of mathematics given by Zermelo Fraenkel Set Theory with the Axiom of Choice (ZFC). Precisely, ZFC is equiconsistent with ETCS augmented with the axiom schema of replacement. Lawvere also called for a similar axiomatization of the two-dimensional structure of categories, functors, and natural transformations. In this talk, I will describe a characterisation of 2-categories of categories internal to a model of ETCS. The resulting theory is the elementary theory of the 2-category of small categories (ET2CSC) of the title. The main result is that the 2-categories of models of ETCS and ET2CSC are biequivalent. The advantage of this approach is that the two-dimensional setting supports a convenient way of incorporating the axiom schema of replacement, albeit in a non-elementary way. This talk is based on joint work with Adrian Miranda.
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Abstract: Lawvere’s Elementary Theory of the Category of Sets (ETCS) provides a set theory which axiomatises the properties of function composition rather than those of a global set membership relation. It provides an important fragment of a category-theoretic foundation of mathematics but is strictly weaker than the traditional foundation of mathematics given by Zermelo Fraenkel Set Theory with the Axiom of Choice (ZFC). Precisely, ZFC is equiconsistent with ETCS augmented with the axiom schema of replacement. In this talk, I will motivate a 2-dimensional version of ETCS which axiomatises the properties of functors and functor composition; this is the elementary theory of the 2-category of small categories (ET2CSC) of the title. The advantage of this approach is that the two-dimensional setting supports a convenient way of incorporating the axiom schema of replacement, albeit in a non-elementary way. This talk is based on joint work with Adrian Miranda.
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Known models of Martin-Löf Type Theory (MLTT) include isofibrations in the category of groupoids and Kan fibrations in the category of simplicial sets. It is an open problem to prove constructively that Kan fibrations model Homotopy Type Theory. One suggested way to approach this problem is with an algebraic perspective; the idea being that by keeping track of the algebraic data throughout calculations, proofs become more constructive. Classically, normal isofibrations are the algebras for the right class of a type theoretic algebraic weak factorisation system which form part of an algebraic model structure. Constructively, however, this link breaks down as this algebraic model structure fails to exist.
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Lawvere’s Elementary Theory of the Category of Sets (ETCS) posits that the category Set is a well-pointed elementary topos with natural numbers object satisfying the axiom of choice. This provides a category theoretic foundation for mathematics which axiomatises the properties of function composition in contrast to Zermelo-Fraenkel set theory with the axiom of choice (ZFC), which axiomatises sets and their membership relation. Furthermore, ETCS augmented with the axiom schema of replacement can be shown to be equiconsistent with ZFC.
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Elementary toposes axiomatise the categorical properties of the category of sets. Their internal logic recovers most of ZF set theory, but it is not possible to formulate statements for which the axiom of replacement is used. Joyal and Moerdijk’s Class Categories were designed to instead axiomatise the categorical property of the category of classes, in which statements including replacement can be formulated. In this talk, I will present Simpson’s minimal axioms for a class category and suggest a 2-categorical version of this, presenting some early results in this direction. Examples of this include CAT and 2-categories of categories— an internal characterisation of CAT based on the elementary theory of the 2-categories of categories. This is joint work in progress with Adrian Miranda.
Undergraduate course, University of Manchester, Mathematics Department, 2023
I lead tutorials for the first year Analysis module. This class had 24 students.
Undergraduate course, University of Manchester, Mathematics Department, 2023
For the first half of the semester, I helped out in problems sessions for third year mathematics students and second year mathematics and physics joint honour students. The class has 80 students. In the second half of the semester, I helped out in problems sessions for third year mathematics students in Complex analysis and Applications. This class has around 30 students.
Undergraduate course, University of Manchester, Mathematics Department, 2024
The GTA work in this module involves helping out groups of second year students with their independent research projects.
Undergraduate course, University of Manchester, Mathematics Department, 2024
Undergraduate course, University of Manchester, Mathematics Department, 2024
This module teaches aerospace engineering students some key mathematical concepts: partial differentiation, line integrals, approximation techniques such as Newton-Raphson and the trapezium rule, ODEs and PDEs.