Alex Corner

Category Theory

My current interests in category theory lie broadly around the topics of monoidal categories, operads, and higher categories.

An ongoing project is working together with Eugenia Cheng to investigate semi-strict higher categories through various iterations of internalisation and enrichment. The papers below discuss this with regard to semi-strict tricategories where the weakness lies in the vertical composition and our original motivation in this work was to use this as a stepping stone on the way to investigating doubly-degenerate Trimble-like 3-categories.

Another longstanding project is working with Nick Gurski to tie up work which we produced during my PhD, together with a further preprint of Gurski's, based around the central theme of action operads and operads with general groups of equivariance. This is submitted and currently awaiting review.

I have picked up various further interests by virtue of having moved into a computer science group. Amongst them include various mathematical aspects of computing and cryptography, despite being a complete non-expert in either of these topics. These interests mostly arise in trying to use my understanding of category theory to put those topics into perspective both for my teaching and other research directions.

Publications

1. Cheng, E., & Corner, A. S. (2024). Weak vertical composition II: totalities. Theory and Applications of Categories, 41(31), 995-1043.

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.

2. Cheng, E., & Corner, A. S. (2024). Weak vertical composition. Theory and Applications of Categories, 42(10), 222-262.

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 \(B\).

3. Corner, A. S. (2019). A universal characterisation of codescent objects. Theory and Applications of Categories, 34(24), 684-713.

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

1. Cheng, E., & Corner, A. S. A higher-order Eckmann-Hilton argument. arXiv: 2606.12357

We give a higher-order higher-dimensional Eckmann-Hilton argument that is entirely algebraic. First we give an explicit argument showing that if we have two monoidal structures on a category with suitable interchange, we can derive a braiding on either of the monoidal structures. Then we show that given a third monoidal structure, with suitable pairwise interchange on any pair of monoidal structures, each canonical braiding is forced to be a symmetry. As a motivating example, we show that for \(n \geq 3\) any \(n\)-degenerate semi-strict \((n+1)\)-category has three suitably coherent monoidal structures on its single hom-category, thus the hom-category has the structure of a symmetric monoidal category.

2. Corner, A. S., & Gurski, N. Operads and equivariance. arXiv: 2603.19854

Operads were originally defined by May to have right actions of the symmetric groups, but later formulations have also used no groups actions at all or group actions by such families as the braid groups. We call such families action operads, as they are the algebraic objects that encode parametrized group actions on operads. In Part I of this paper, we study the basic algebra of action operads \(\Lambda\) and the \(\Lambda\)-operads they act upon. In Part II, we study \(\Lambda\)-operads in the 2-category of small categories.

3. Corner, A. S., & Gurski, N. Operads with general groups of equivariance, and some 2-categorical aspects of operads in \(\mathbf{Cat}\). arXiv: 1312.5910

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

1. Corner, A. S. Day convolution for monoidal bicategories.

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

1. PSSL 110 Talk - Operads, Equivariance, and Pseudo-Commutativity.

This talk was contributed to PSSL 110, hosted by TalTech in the Old Town of Tallinn. Note: The claim about the operad of braid groups having a pseudo-commutative structure appears to be false and is noted in the arXiv preprint 2604.19854.

2. CT2024 Talk - A higher-dimensional Eckmann–Hilton argument.

This plenary talk was presented at the International Category Theory Conference at Universidade de Santiago de Compostela.

3. Algebraic Topology Seminar Talk - Weak vertical composition.

This talk was presented at the Algebraic Topology Seminar at the University of Sheffield.

4. CT2023 Poster - Weak vertical composition: totalities.

This poster was presented at the International Category Theory Conference at UCLouvain in Louvain-la-Neuve.

5. Blog Post - Codescent Objects and Coherence

This blog post was written as part of the first Kan Extension Seminar organised by Emily Riehl.

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