Adrian Miranda

Adrian Miranda

Australian Category theorist with an interest in logic

Research

Abstract Kleisli structures on $2$-categories

Published in Electronic Proceedings in Theoretical Computer Science, to appear., 2024

Fuhrmann introduced Abstract Kleisli structures to model call-by-value programming languages with side effects, and showed that they correspond to monads satisfying a certain equalising condition on the unit. We first extend this theory to non-strict morphisms of monads, and to incorporate $2$-cells of monads. We then further extend this to a theory of abstract Kleisli structures on $2$-categories, characterising when the original pseudomonad can be recovered by the abstract Kleisli structure on its $2$-category of free-pseudoalgebras.

Recommended citation: Miranda, Adrian. (2024). https://arxiv.org/abs/2404.00900

The elementary theory of the 2-category of small categories

Published in Theory and Applications of Categories, to appear., 2024

We give an elementary description of $2$-categories $\mathbf{Cat}(\mathcal{E})$ of internal categories, functors and natural transformations, where $\mathcal{E}$ is a category modelling Lawveres elementary theory of the category of sets (ETCS). This extends Bourkes characterisation of $2$-categories $\mathbf{Cat}(\mathcal{E})$ where $\mathcal{E}$ has pullbacks to take account for the extra properties in ETCS, and Lawveres 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 $\mathbf{Cat}(\mathcal{E})$, results which we believe are of independent interest.

Recommended citation: Hughes, C., Miranda, A. (2024). https://arxiv.org/abs/2403.03647

Eilenberg-Moore Bicategories for Opmonoidal Pseudomonads

Published in Awaiting peer review, 2024

We analyse compatibility between monads and monoidal structures in the two-dimensional setting. We describe sufficient conditions for monoidal structures to lift to the Eilenberg-Moore pseudoalgebras. We then extend these results to braids, syllapses and symmetries. To achieve these results we define the $\mathbf{Gray}$-tensor product of pseudomonads, and examine its interaction with the Eilenberg-Moore construction.

Recommended citation: Miranda, Adrian. (2024). https://arxiv.org/abs/2402.11703

Enriched Kleisli objects for pseudomonads

Published in Awaiting peer review, 2023

A pseudomonad on a 2-category whose underlying endomorphism is a 2-functor can be seen as a diagram $\mathbf{Psmnd} \rightarrow \mathbf{Gray}$ for which weighted limits and colimits can be considered. The 2-category of pseudoalgebras, pseudomorphisms and 2-cells is such a $\mathbf{Gray}$-enriched weighted limit, however neither the Kleisli bicategory nor the 2-category of free pseudoalgebras are the analogous weighted colimit. In this paper we describe the actual weighted colimit via a presentation, and show that the comparison 2-functor induced by any other pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a consequence, we see that biessential surjectivity on objects characterises left pseudoadjoints whose codomains have an up to biequivalence version of the universal property for Kleisli objects. This motivates a homotopical study of Kleisli objects for pseudomonads, and to this end we show that the weight for Kleisli objects is cofibrant in the projective model structure on the $\mathbf{Gray}$-category of enriched presheaves on $\mathbf{Psmnd}$.

Recommended citation: Miranda, Adrian. (2023). https://arxiv.org/abs/2311.15618

A semi-strictly generated closed structure on $Gray$-categories

Published in Journal of Pure and Applied Algebra, to appear., 2023

We show that the semi-strictly generated internal homs of $\mathbf{Gray}$-categories defined in a previous paper underlie a closed structure on the category $\mathbf{Gray}$-$\mathbf{Cat}$ of $\mathbf{Gray}$-categories and $\mathbf{Gray}$-functors. The morphisms of the internal hom are composites of those trinatural transformations which satisfy the unit and composition conditions for pseudonatural transformations on the nose rather than up to an invertible $3$-cell. Such trinatural transformations leverage three-dimensional strictification while overcoming the challenges posed by failure of middle four interchange to hold in $\mathbf{Gray}$-categories. As a result we obtain a closed structure that is only partially monoidal with respect to the Crans monoidal structure. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories, which will be improved further in a forthcoming paper.

Recommended citation: Miranda, Adrian. (2023). https://arxiv.org/abs/2311.11403

Strictifying Operational Coherences and Weak Functor Classifiers in Low Dimensions

Published in Awaiting peer review, 2023

Weak structures abound in higher category theory, but are often suitably equivalent to stricter structures that are easier to understand. We extend strictification for tricategories and trihomomorphisms to trinatural transformations, trimodifications and perturbations. Along the way we distinguish between the operational coherences, which are possible to strictify, and the coherences on globular inputs, which remain weak. We introduce generalised path objects for $\mathbf{Gray}$-categories, which help reduce proofs in the three-dimensional setting to known results. Upon closing the resulting semi-strict trinatural transformations under composition, we state the hom-triequivalences of what we expect to be a semi-strictification tetra-adjunction.

Recommended citation: Miranda, Adrian. (2023). https://arxiv.org/abs/2307.01498

Topics in Low Dimensional Higher Category Theory

Published in Ph. D. Thesis, Macquarie University, 2023

The first part of this thesis concerns strictification in low-dimensional higher category theory. We review existing theory in dimensions two and three, and extend some results which are known in the two dimensional setting to the three dimensional setting, obtaining new results there. New theory includes a notion of semi-strictness for trinatural transformations, a relationship which can reasonably be called a ‘semi-strictification tetra-adjunction’, and a closed structure [−, ?] on Gray-Cat which features trinatural transformations that admit some decomposition into semi-stricts. Following this, we consider enrichment over (Gray-Cat, [−, ?] ) and the relation to tetracategories. Free constructions, and a distinction between operational and free coherence data, simplify and guide our study of semi-strictification in low dimensions. Freeness is needed to leverage semi-strictification adjunctions of (n − 1)-dimensional data towards semi-strictification for n-dimensional categories. We provide precise conjectures for semi-strict models keeping interchange weak in dimensions four and five, for which we are able to provide strong evidence and outline a promising proof strategy in dimension four. The second part of this thesis studies generalisations of the Kleisli construction to the two-dimensional setting, and the broader theory of tricategorical limits and colimits. Main examples such as Gray and Hom, the tricategory of bicategories and weak (2, k)-transfors, will be shown to have trilimits and tricolimits. Tricategorical notions are related to notions enriched over Gray as a monoidal model category, in analogy to the bicategorical setting and overcoming the challenges presented by non-cofibrant 2-categories. A particular related pair of colimits is examined. These categorify the usual Kleisli construction for monads to the level of pseudomonads, one in the stricter Gray-categorical sense and the other in the weaker tricategorical sense. The weaker ‘trikleisli’ left pseudoadjoints in Hom are precisely the biessentially surjective on objects ones, and use the fact that biessentially surjective on objects pseudofunctors are closed under composition to describe what we conjecture to be the free cocompletion of a tricategory under trikleisli objects for pseudomonads. This extends the theory of wreaths to the pseudomonad setting and has potential applications to pseudo double categories and enrichment over monoidal bicategories. We explore the first of these applications in detail, using this perspective to categorify the notion of cofunctor and natural cotransformation to the pseudo double category setting, and describe a strictification of double pseudocofunctors. We extend these ideas to a definition of pseudo triple category and outline a program aimed at strictifying them using a slice-wise version of B¨ohm’s tensor product of double categories. As a different direction in our study of the formal theory of pseudomonads, we extend the theory of abstract Kleisli structures to the pseudomonad setting and show that these correspond to pseudomonads whose units satisfy a bicategorical limit condition.

Recommended citation: Miranda, Adrian. (2022). https://figshare.mq.edu.au/articles/thesis/Topics_in_low_dimensional_higher_category_theory/24872088/1

What is the universal property of the $2$-category of monads?

Published in Theory and Applications of Categories, to appear., 2022

For a 2-category $\mathcal{K}$, we consider Streets 2-category $\mathbf{Mnd}(\mathcal{K})$ of monads in $\mathcal{K}$, along with Lack and Streets $2$-category $\mathbf{EM}(\mathcal{K})$ and the identity-on-objects-and-$1$-cells $2$-functor $\mathbf{Mnd}(\mathcal{K}) → \mathbf{EM}(\mathcal{K})$ between them. We show that this $2$-functor can be obtained as a free completion of the 2-functor $1:\mathcal{K}→\mathcal{K}$. We do this by regarding $2$-functors which act as the identity on both objects and $1$-cells as categories enriched a cartesian closed category BO whose objects are identity-on-objects functors. We also develop some of the theory of BO-enriched categories.

Recommended citation: Lack, Stephen., Miranda, Adrian. (2022). https://arxiv.org/abs/2211.02210

Internal Categories

Published in Masters Thesis, Macquarie University, 2018

In our first chapter, we will define categories, functors, and natural transformations internally to any category with pullbacks ε, and we will prove in detail that they form a Cat-enriched category, or 2-category Cat (ε), with powers by 2 and any conical limits that ε also has. Along the way we will describe how certain familiar notions of category theory can be made sense of internally. In Chapter Two we will explore how some properties of ε are inherited by, or give rise to other properties in Cat (ε). In Chapter Three we will investigate the extension of the assignment ε -> Cat (ε) to various 2-functors, and in particular equip one of them with various monad-like structures. One of these was remarked upon in [6], but to our knowledge the other two have not appeared elsewhere in the literature. Chapter Four will be an intermezzo on the Grothendieck Construction in preparation for Chapter Five, where we will explore factorisations of internal functors, including in particular the comprehensive factorisation.

Recommended citation: Miranda, Adrian. (2018). https://figshare.mq.edu.au/articles/thesis/Internal_categories/19434626/1