Searcharxiv⌕ Search

arXiv subjects

Stephen Lack

Publications and source records attributed to Stephen Lack.

At least 19 recordsLinked to original sources
arXiv2025

Nerves of generalized multicategories

For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction defines a fully faithful functor from the category $\mathbf{Cat}_T({\mathcal E})$ of $T$-categories to the category $s_T({\mathcal E})$ of $T$-simplicial objects, whose essential image is characterized by a simple condition. We show that the category $s_T({\mathcal E})$ is enriched over the category of simplicial sets, and that this induces the usual 2-category structure on $\mathbf{Cat}_T({\mathcal E})$. We also study enriched limits and colimits in $s_T({\mathcal E})$ and $\mathbf{Cat}_T({\mathcal E})$, and show that if ${\mathcal E}$ is locally finitely presentable and $T$ is finitary, then $\mathbf{Cat}_T({\mathcal E})$ is locally finitely presentable as a 2-category and $s_T({\mathcal E})$ is locally finitely presentable as a simplicially-enriched category.

math.CT↗
arXiv2025

The familial nature of enrichment over virtual double categories

Originally enriched categories were defined over a monoidal category, but it was gradually realized that important examples can only be included when one enriches over more general structures such as bicategories and virtual double categories. We show that, as well as allowing more examples, working over virtual double categories also gives better formal properties. We study the 2-functor sending a virtual double category to the 2-category of categories enriched over it. We show that this is a parametric right 2-adjoint, and in fact is familial. We also show how a ``families construction'' for virtual double categories can be used to give a formal construction of the 2-category of categories enriched over a virtual double category.

math.CT↗
arXiv2024

A characterisation for the category of Hilbert spaces

The categories of real and of complex Hilbert spaces with bounded linear maps have received purely categorical characterisations by Chris Heunen and Andre Kornell. These characterisations are achieved through Sol\`er's theorem, a result which shows that certain orthomodularity conditions on a Hermitian space over an involutive division ring result in a Hilbert space with the division ring being either the reals, complexes or quaternions. The characterisation by Heunen and Kornell makes use of a monoidal structure, which in turn excludes the category of quaternionic Hilbert spaces. We provide an alternative characterisation without the assumption of monoidal structure on the category. This new approach not only gives a new characterisation of the categories of real and of complex Hilbert spaces, but also the category of quaternionic Hilbert spaces.

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

Accessible categories with a class of limits

In this paper we characterize those accessible $\mathcal V$-categories that have limits of a specified class. We do this by introducing the notion of companion $\mathfrak C$ for a class of weights $\Psi$, as a collection of special types of colimit diagrams that are compatible with $\Psi$. We then characterize the accessible $\mathcal V$-categories with $\Psi$-limits as those accessibly embedded and $\mathfrak C$-virtually reflective in a presheaf $\mathcal V$-category, and as the $\mathcal V$-categories of $\mathfrak C$-models of sketches. This allows us to recover the standard theorems for locally presentable, locally multipresentable, and locally polypresentable categories as instances of the same general framework. In addition, our theorem covers the case of any weakly sound class $\Psi$, and provides a new perspective on the case of weakly locally presentable categories.

math.CT↗
arXiv2022

The oplax limit of an enriched category

We show that 2-categories of the form $\mathscr{B}\mbox{-}\mathbf{Cat}$ are closed under slicing, provided that we allow $\mathscr{B}$ to range over bicategories (rather than, say, monoidal categories). That is, for any $\mathscr{B}$-category $\mathbb{X}$, we define a bicategory $\mathscr{B}/\mathbb{X}$ such that $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$. The bicategory $\mathscr{B}/\mathbb{X}$ is characterized as the oplax limit of $\mathbb{X}$, regarded as a lax functor from a chaotic category to $\mathscr{B}$, in the 2-category $\mathbf{BICAT}$ of bicategories, lax functors and icons. We prove this conceptually, through limit-preservation properties of the 2-functor $\mathbf{BICAT}\to 2\mbox{-}\mathbf{CAT}$ which maps each bicategory $\mathscr{B}$ to the 2-category $\mathscr{B}\mbox{-}\mathbf{Cat}$. When $\mathscr{B}$ satisfies a mild local completeness condition, we also show that the isomorphism $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$ restricts to a correspondence between fibrations in $\mathscr{B}\mbox{-}\mathbf{Cat}$ over $\mathbb{X}$ on the one hand, and $\mathscr{B}/\mathbb{X}$-categories admitting certain powers on the other.

math.CT↗
arXiv2022

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

For a 2-category $\mathcal{K}$, we consider Street's 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street's 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$) $\to$ EM($\mathcal{K}$) between them. We show that this 2-functor can be obtained as a "free completion" of the 2-functor $1\colon \mathcal{K} \to \mathcal{K}$. 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 $\mathbf{BO}$ whose objects are identity-on-objects functors. We also develop some of the theory of $\mathbf{BO}$-enriched categories.

math.CT↗
arXiv2022

Virtual concepts in the theory of accessible categories

We provide a new characterization of enriched accessible categories by introducing the two new notions of virtual reflectivity and virtual orthogonality as a generalization of the usual reflectivity and orthogonality conditions for locally presentable categories. The word virtual refers to the fact that the reflectivity and orthogonality conditions are given in the free completion of the $\mathcal V$-category involved under small limits, instead of the $\mathcal V$-category itself. In this way we hope to provide a clearer understanding of the theory as well as a useful way of recognizing accessible $\mathcal V$-categories. In the last section we prove that the 2-category of accessible $\mathcal V$-categories, accessible $\mathcal V$-functors, and $\mathcal V$-natural transformations has all flexible limits.

math.CT↗
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↗
arXiv2021

Flat vs. filtered colimits in the enriched context

The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as $\mathbf{Ab},\mathbf{SSet},\mathbf{Cat}$ and $\mathbf{Met}$. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: (1) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow$ (b) holds; (2) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow $ (b) holds up to Cauchy completion; (3) explore some examples not covered by (1) or (2).

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↗
arXiv2019

Enriched Regular Theories

Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosick\'y, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Considering a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosick\'y.

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

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

Operadic categories and their skew monoidal categories of collections

I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category S of finite sets and functions, for which an operad is just a symmetric operad in the usual sense, the first skew monoidal category has underlying category [N, Set], and the second is the usual monoidal category of collections [P, Set] with the substitution monoidal structure.

math.CT↗
arXiv2016

Weak multiplier bimonoids

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided monoidal categories. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

math.CT↗
arXiv2015

A simplicial approach to multiplier bimonoids

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.

math.CT↗