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Richard Garner

Publications and source records attributed to Richard Garner.

At least 37 records · Page 2Linked to original sources
arXiv2020

Generalising the étale groupoid--complete pseudogroup correspondence

We prove a generalisation of the correspondence, due to Resende and Lawson--Lenz, between étale groupoids---which are topological groupoids whose source map is a local homeomorphisms---and complete pseudogroups---which are inverse monoids equipped with a particularly nice representation on a topological space. Our generalisation improves on the existing functorial correspondence in four ways. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from étale groupoids to what we call partite étale groupoids. Thirdly, we generalise from étale groupoids to source-étale categories, and on the other side, from inverse monoids to restriction monoids. Fourthly, and most far-reachingly, we generalise from topological étale groupoids to étale groupoids internal to any join restriction category C with local glueings; and on the other side, from complete pseudogroups to ``complete C-pseudogroups'', i.e., inverse monoids with a nice representation on an object of C. Taken together, our results yield an equivalence, for a join restriction category C with local glueings, between join restriction categories with a well-behaved functor to C, and partite source-étale internal categories in C. In fact, we obtain this by cutting down a larger adjunction between arbitrary restriction categories over C, and partite internal categories in C. Beyond proving this main result, numerous applications are given, which reconstruct and extend existing correspondences in the literature, and provide general formulations of completion processes.

math.CT↗
arXiv2019

Inner automorphisms of groupoids

Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism of $H$. This leads naturally to a definition of "inner automorphism" applicable to the objects of any category. Bergman and Hofstra--Parker--Scott have computed these inner automorphisms for various structures including $k$-algebras, monoids, lattices, unital rings, and quandles---showing that, in each case, they are given by an obvious notion of conjugation. In this note, we compute the inner automorphisms of groupoids, showing that they are exactly the automorphisms induced by conjugation by a bisection. The twist is that this result is false in the category of groupoids and homomorphisms; to make it true, we must instead work with the less familiar category of groupoids and comorphisms in the sense of Higgins and Mackenzie. Besides our main result, we also discuss generalisations to topological and Lie groupoids, to categories and to partial automorphisms, and examine the link with the theory of inverse semigroups.

math.CT↗
arXiv2018

Operadic categories and décalage

Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres" -- also objects of the same category -- subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the décalage comonad D on small categories. A simple case involves unary operadic categories -- ones wherein each map has exactly one abstract fibre -- which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad induced on the category of D-coalgebras by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a "modified décalage" comonad on the arrow category of Cat.

math.CT↗
arXiv2018

Lifting accessible model structures

A Quillen model structure is presented by an interacting pair of weak factorization systems. We prove that in the world of locally presentable categories, any weak factorization system with accessible functorial factorizations can be lifted along either a left or a right adjoint. It follows that accessible model structures on locally presentable categories - ones admitting accessible functorial factorizations, a class that includes all combinatorial model structures but others besides - can be lifted along either a left or a right adjoint if and only if an essential "acyclicity" condition holds. A similar result was claimed in a paper of Hess-Kedziorek-Riehl-Shipley, but the proof given there was incorrect. In this note, we explain this error and give a correction, and also provide a new statement and a different proof of the theorem which is more tractable for homotopy-theoretic applications.

math.CT↗
arXiv2018

An enriched view on the extended finitary monad--Lawvere theory correspondence

We give a new account of the correspondence, first established by Nishizawa--Power, between finitary monads and Lawvere theories over an arbitrary locally finitely presentable base. Our account explains this correspondence in terms of enriched category theory: the passage from a finitary monad to the corresponding Lawvere theory is exhibited as an instance of free completion of an enriched category under a class of absolute colimits. This extends work of the first author, who established the result in the special case of finitary monads and Lawvere theories over the category of sets; a novel aspect of the generalisation is its use of enrichment over a bicategory, rather than a monoidal category, in order to capture the monad--theory correspondence over all locally finitely presentable bases simultaneously.

math.CT↗
arXiv2017

Shapely monads and analytic functors

In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that "any two operations of the same shape agree". An important part of this work is the study of analytic functors between presheaf categories, which are a common generalisation of Joyal's analytic endofunctors on sets and of the parametric right adjoint functors on presheaf categories introduced by Diers and studied by Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among the analytic endofunctors, and may be characterised as the submonads of a universal analytic monad with "exactly one operation of each shape". In fact, shapeliness also gives a way to define the data and axioms of a structure directly from its graphical calculus, by generating a free shapely monad on the basic operations of the calculus. In this paper we do this for some of the examples listed above; in future work, we intend to do so for graphical calculi such as Milner's bigraphs, Lafont's interaction nets, or Girard's multiplicative proof nets, thereby obtaining canonical notions of denotational model.

cs.LO↗
arXiv2016

Cocompletion of restriction categories

Restriction categories were introduced as a way of generalising the notion of partial map categories. In this paper, we define cocomplete restriction category, and give the free cocompletion of a small restriction category as a suitably defined category of restriction presheaves. We also consider the case where our restriction category is locally small.

math.CT↗
arXiv2016

When coproducts are biproducts

Among right-closed monoidal categories with finite coproducts, we characterise those with finite biproducts as being precisely those in which the initial object and the coproduct of the unit with itself admit right duals. This generalises Houston's result that any compact closed category with finite coproducts admits biproducts.

math.CT↗
arXiv2015

Enriched categories as a free cocompletion

This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory -- categorifying the classical theory of categories enriched in a monoidal category -- up to a description of the free cocompletion of an enriched bicategory under a class of weighted bicolimits. The second objective is to describe a universal property of the process assigning to a monoidal category V the equipment of V-enriched categories, functors, transformations, and modules; we do so by considering, more generally, the assignation sending an equipment C to the equipment of C-enriched categories, functors, transformations, and modules, and exhibiting this as the free cocompletion of a certain kind of enriched bicategory under a certain class of weighted bicolimits.

math.CT↗
arXiv2015

Hochschild homology, lax codescent, and duplicial structure

We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Bohm and Stefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.

math.CT↗
arXiv2015

Coalgebras governing both weighted Hurwitz products and their pointwise transforms

We give further insights into the weighted Hurwitz product and the weighted tensor product of Joyal species. Our first group of results relate the Hurwitz product to the pointwise product, including the interaction with Rota--Baxter operators. Our second group of results explain the first in terms of convolution with suitable bialgebras, and show that these bialgebras are in fact obtained in a particularly straightforward way by freely generating from pointed coalgebras. Our third group of results extend this from linear algebra to two-dimensional linear algebra, deriving the existence of weighted Hurwitz monoidal structures on the category of species using convolution with freely generated bimonoidales. Our final group of results relate Hurwitz monoidal structures with equivalences of of Dold--Kan type.

math.CT↗
arXiv2015

Commutativity

We describe a general framework for notions of commutativity based on enriched category theory. We extend Eilenberg and Kelly's tensor product for categories enriched over a symmetric monoidal base to a tensor product for categories enriched over a normal duoidal category; using this, we re-find notions such as the commutativity of a finitary algebraic theory or a strong monad, the commuting tensor product of two theories, and the Boardman-Vogt tensor product of symmetric operads.

math.CT↗
arXiv2015

Algebraic weak factorisation systems I: accessible AWFS

Algebraic weak factorisation systems (AWFS) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad--monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of AWFS---drawing on work of previous authors---and complete the theory with two main new results. The first provides a characterisation of AWFS and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an AWFS by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an AWFS, and that the AWFS so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of AWFS are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.

math.CT↗
arXiv2015

Algebraic weak factorisation systems II: categories of weak maps

We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of spans with left leg an acyclic fibration. We moreover show that the 2-functor sending each AWFS on a suitable category to its cofibrant replacement comonad has a fully faithful right adjoint: so exhibiting the theory of comonads, and dually of monads, as incorporated into the theory of AWFS. We also describe various applications of the general theory: to the generalised sketches of Kinoshita-Power-Takeyama, to the two-dimensional monad theory of Blackwell-Kelly-Power, and to the theory of dg-categories.

math.CT↗
arXiv2015

Orientals and cubes, inductively

We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular omega-category.

math.CT↗
arXiv2014

The Catalan simplicial set

The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.

math.CT↗
arXiv2014

The Isbell monad

In 1966, John Isbell introduced a construction on categories which he termed the "couple category" but which has since come to be known as the Isbell envelope. The Isbell envelope, which combines the ideas of contravariant and covariant presheaves, has found applications in category theory, logic, and differential geometry. We clarify its meaning by exhibiting the assignation sending a locally small category to its Isbell envelope as the action on objects of a pseudomonad on the 2-category of locally small categories; this is the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell monad as categories equipped with a cylinder factorisation system; this notion, which appears to be new, is an extension of Freyd and Kelly's notion of factorisation system from orthogonal classes of arrows to orthogonal classes of cocones and cones.

math.CT↗