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John Bourke

Publications and source records attributed to John Bourke.

At least 19 recordsLinked to original sources
arXiv2025

A constructive approach to the double-categorical small object argument

Bourke and Garner described how to cofibrantly generate algebraic weak factorisation systems by a small double category of morphisms. However they did not give an explicit construction of the resulting factorisations as in the classical small object argument. In this paper we give such an explicit construction, as the colimit of a chain, which makes the result applicable in constructive settings; in particular, our methods provide a constructive proof that the effective Kan fibrations introduced by Van den Berg and Faber appear as the right class of an algebraic weak factorisation system.

math.CT↗
arXiv2024

Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation

Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal fibration is - crucially - a strict monoidal functor, rather than a pseudo or lax monoidal functor. Other examples include monoidal double categories, double fibrations, and intercategories. We provide an explanation for this phenomenon from the perspective of enhanced 2-categories, which are 2-categories having a distinguished subclass of 1-cells representing the strict morphisms. As part of our development, we introduce enhanced 2-categorical limit sketches and explain how this setting addresses shortcomings in the theory of 2-categorical limit sketches. In particular, we establish the symmetry of internalisation for such structures, entailing, for instance, that a monoidal double category is equivalently a pseudomonoid in an enhanced 2-category of double categories, or a pseudocategory in an enhanced 2-category of monoidal categories.

math.CT↗
arXiv2023

On $2$-categorical $\infty$-cosmoi

Recently Riehl and Verity have introduced $\infty$-cosmoi, which are certain simplicially enriched categories with additional structure. In this paper we investigate those $\infty$-cosmoi which are in fact $2$-categories; we shall refer to these as $2$-cosmoi. We show that each $2$-category with flexible limits gives rise to a $2$-cosmos whose distinguished class of isofibrations consists of the normal isofibrations. Many examples arise in this way, and we show that such $2$-cosmoi are minimal as Cauchy-complete $2$-cosmoi. Finally, we investigate accessible $2$-cosmoi and develop a few aspects of their basic theory.

math.CT↗
arXiv2022

A skew approach to enrichment for Gray-categories

It is well known that the category of Gray-categories does not admit a monoidal biclosed structure that models weak higher-dimensional transformations. In this paper, the first of a series on the topic, we describe several skew monoidal closed structures on the category of Gray-categories, one of which captures higher lax transformations, and another which models higher pseudo-transformations.

math.CT↗
arXiv2022

The Ages of Galactic Bulge Stars with Realistic Uncertainties

Using modern isochrones with customized physics and carefully considered statistical techniques, we recompute the age distribution for a sample of 91 micro-lensed dwarfs in the Galactic bulge presented by Bensby et al. (2017) and do not produce an age distribution consistent with their results. In particular, our analysis finds that only 15 of 91 stars have ages younger than 7 Gyr, compared to their finding of 42 young stars in the same sample. While we do not find a constituency of very young stars, our results do suggest the presence of an $\sim8$ Gyr population at the highest metallicities, thus contributing to long-standing debate about the age--metallicity distribution of the Galactic bulge. We supplement this with attempts at independent age determinations from two sources of photometry, BDBS and \textit{Gaia}, but find that the imprecision of photometric measurements prevents reliable age and age uncertainty determinations. Lastly, we present age uncertainties derived using a first-order consideration of global modeling uncertainties in addition to standard observational uncertainties. The theoretical uncertainties are based on the known variance of free parameters in the 1D stellar evolution models used to generate isochrones, and when included, result in age uncertainties of $2$--$5$ Gyr for this spectroscopically well-constrained sample. These error bars, which are roughly twice as large as typical literature values, constitute realistic lower limits on the true age uncertainties.

astro-ph.SR↗
arXiv2021

Accessible $\infty$-cosmoi

We introduce the notion of an accessible $\infty$-cosmos and prove that these include the basic examples of $\infty$-cosmoi and are stable under the main constructions. A consequence is that the vast majority of known examples of $\infty$-cosmoi are accessible. By the adjoint functor theorem for homotopically enriched categories which we proved in an earlier paper, joint with Lukas Vokrinek, it follows, for instance, that all such $\infty$-cosmoi have flexibly weighted homotopy colimits.

math.CT↗
arXiv2020

Adjoint functor theorems for homotopically enriched categories

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched version of Freyd's adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour - in particular, when $\mathcal V$ is the category of simplical sets we obtain a homotopical adjoint functor theorem appropriate to the $\infty$-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched setting, in particular obtaining homotopical cocompleteness results for accessible $\infty$-cosmoi.

math.CT↗
arXiv2020

Algebraically cofibrant and fibrant objects revisited

We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent transferred weak model structures on both the categories of algebraically cofibrant and algebraically fibrant objects. Under additional assumptions, these transferred weak model structures are shown to be left, right or Quillen model structures. By combining both constructions, we show that each combinatorial weak model category is connected, via a chain of Quillen equivalences, to a combinatorial Quillen model category in which all objects are fibrant.

math.CT↗
arXiv2020

Accessible aspects of 2-category theory

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the structure of monoidal, but not strict monoidal, categories) then the 2-category in question is accessible. Furthermore, we explore the flexible limits that such 2-categories possess and their interaction with filtered colimits.

math.CT↗
arXiv2018

Iterated algebraic injectivity and the faithfulness conjecture

Algebraic injectivity was introduced to capture homotopical structures like algebraic Kan complexes. But at a much simpler level, it allows one to describe sets with operations subject to no equations. If one wishes to add equations (or operations of greater complexity) then it is natural to consider iterated algebraic injectives, which we introduce and study in the present paper. Our main application concerns Grothendieck's weak $\omega$-groupoids, introduced in Pursuing Stacks, and the closely related definition of weak $\omega$-category due to Maltsiniotis. Using $\omega$ iterations we describe these as iterated algebraic injectives and, via this correspondence, prove the faithfulness conjecture of Maltsiniotis. Through work of Ara, this implies a tight correspondence between the weak $\omega$-categories of Maltsiniotis and those of Batanin/Leinster.

math.CT↗
arXiv2018

Monads and theories

Given a locally presentable enriched category $\mathcal{E}$ together with a small dense full subcategory $\mathcal A$ of arities, we study the relationship between monads on $\mathcal E$ and identity-on-objects functors out of $\mathcal A$, which we call $\mathcal A$-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised as the $\mathcal A$-nervous monads---those for which the conclusions of Weber's nerve theorem hold---and the $\mathcal A$-theories, which we introduce here. The resulting equivalence between $\mathcal A$-nervous monads and $\mathcal A$-theories is best possible in a precise sense, and extends almost all previously known monad--theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak $\omega$-groupoids. Besides establishing our general correspondence and illustrating its reach, we study good properties of $\mathcal A$-nervous monads and $\mathcal A$-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Melli\`es and Weber.

math.CT↗
arXiv2017

Braided skew monoidal categories

We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. These braidings are shown to arise from, and classify, cobraidings (also known as coquasitriangular structures) on bialgebras. Using a multicategorical approach we also describe examples of braidings on skew monoidal categories arising from 2-category theory.

math.CT↗
arXiv2017

Equipping weak equivalences with algebraic structure

We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T-algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the weak equivalences in a combinatorial model category can be equipped with algebraic structure.

math.CT↗
arXiv2017

Free skew monoidal categories

In the paper "Triangulations, orientals, and skew monoidal categories", the free monoidal category Fsk on a single generating object was described. We sharpen this by giving a completely explicit description of Fsk, and so of the free skew monoidal category on any category. As an application we describe adjunctions between the operad for skew monoidal categories and various simpler operads. For a particular such operad L, we identify skew monoidal categories with certain colax L-algebras.

math.CT↗
arXiv2016

Note on the construction of globular weak omega-groupoids from types, topological spaces etc

A short introduction to Grothendieck weak omega-groupoids is given. Our aim is to give evidence that, in certain contexts, this simple language is a convenient one for constructing globular weak omega-groupoids. To this end, we give a short reworking of van den Berg and Garner's construction of a Batanin weak omega-groupoid from a type using the language of Grothendieck weak omega-groupoids.

math.CT↗
arXiv2015

Skew structures in 2-category theory and homotopy theory

We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and bicategories. Using the skew framework, we adapt Eilenberg and Kelly's theorem relating monoidal and closed structure to the homotopical setting. This is applied to the construction of monoidal bicategories arising from the pseudo-commutative 2-monads of Hyland and Power.

math.CT↗