arXiv subjects

Alexander S. Corner

Publications and source records attributed to Alexander S. Corner.

5 recordsLinked to original sources

arXiv2026

A higher-order Eckmann-Hilton argument

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 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.

math.CT↗

arXiv2023

Weak vertical composition II: totalities

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.

math.CT↗

arXiv2022

Weak vertical composition

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 $B$.

math.CT↗

arXiv2017

A Universal Characterisation of Codescent Objects

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 extrapseudonatural 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.

math.CT↗

arXiv2014

Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat

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 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.

math.CT↗