For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation ... more For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation of the ‘coarse ’ underlying space together with the local isotropy groups of M. We prove that, for any local system on M, this complex S(M) has the same cohomology as M. The use of S(M) in explicit calculations is illustrated in the example of the ‘teardrop ’ orbifold. Introduction. Orbifolds or V-manifolds were first introduced by Satake [9], and arise naturally in many ways. For example, the orbit space of any proper action by a (discrete) group on a manifold has the structure of an orbifold; this applies in particular to moduli spaces. Furthermore, the orbit space of any almost free action by a
Abstract. Two constructions of paths in double categories are studied, providing algebraic versio... more Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fc-multicategories, with representable identities in the second case.
Abstract. We give two related universal properties of the span construction. The first involves s... more Abstract. We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these “jointed ” oplax morphisms.
We introduce the notion of weakly globular double categories, a particular class of strict double... more We introduce the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories. We show that this model is suitably equivalent to bicategories and give an explicit description of the functors involved in this biequivalence. As an application we show that groupoidal weakly globular double categories model homotopy 2-types.
This paper introduces the construction of a weakly globular double category of fractions for a ca... more This paper introduces the construction of a weakly globular double category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples.
Two constructions of paths in double categories are studied, providing algebraic versions of the ... more Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These con- structions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster's fc-multicategories, with representable identities in the second case.
In this paper we introduce a description of ordered groupoids as a particular type of double cate... more In this paper we introduce a description of ordered groupoids as a particular type of double categories. This enables us to turn Lawson's correspondence between ordered groupoids and left-cancellative categories into a biequivalence. We use this to identify which ordered functors are maps of sites in the sense that they give rise to geometric morphisms between the induced sheaf categories, and establish a Comparison Lemma for maps between Ehresmann sites.
We present two generalizations of the Span construction. The first general- ization gives Span of... more We present two generalizations of the Span construction. The first general- ization gives Span of a category with all pullbacks as a (weak) double category. This dou- ble category SpanA can be viewed as the free double category on the vertical category A where every vertical arrow has both a companion and a conjoint (and these companions and conjoints are adjoint to each other). Thus defined, Span : Cat → Doub becomes a 2-functor, which is a partial left bi-adjoint to the forgetful functor Vrt : Doub → Cat, which sends a double category to its category of vertical arrows. The second generalization gives Span of an arbitrary category as an oplax normal dou- ble category. The universal property can again be given in terms of companions and conjoints and the presence of their composites. Moreover, SpanA is universal with this property in the sense that Span : Cat → OplaxNDoub is left bi-adjoint to the forgetful functor which sends an oplax double category to its vertical arrow category.
We show that the two binary operations in double inverse semigroups, as considered by Kock [2007]... more We show that the two binary operations in double inverse semigroups, as considered by Kock [2007], necessarily coincide.
We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly t... more We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly to a specific class of smooth groupoids. As an application, we construct fibered products of orbifolds and prove a change-of-base formula for sheaf cohomology. Mathematics Subject Classifications (1991): 22A22, 57S15, 55N30, 18B25.
We give two related universal properties of the span construction. The first involves sinister mo... more We give two related universal properties of the span construction. The first involves sinister morphisms out of the base category and sinister transformations. The second involves oplax morphisms out of the bicategory of spans having an extra property; we call these "jointed" oplax morphisms.
In this paper we introduce two notions |systems of brant objects and bration structures| which wi... more In this paper we introduce two notions |systems of brant objects and bration structures| which will allow us to associate to a bicategory a homotopy bicategory Ho(B) in such a way that Ho(B) is the universal way to add pseudo-inverses to weak equivalences inB. Furthermore, Ho(B) is locally small whenB is and Ho(B) is a 2-category whenB is. We thereby resolve two of the problems with known approaches to bicategorical localization. As an important example, we describe a bration structure on the 2-category of prestacks on a site and prove that the resulting homotopy bicategory is the 2-category of stacks. We also show how this example can be restricted to obtain algebraic, dierentiable and topological (respectively) stacks as homotopy categories of algebraic, dierential and
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicat... more We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon cohomology.
The reverse derivative is a fundamental operation in machine learning and automatic differentiati... more The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.
For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation ... more For any orbifold M, we explicitly construct a simplicial complex S(M) from a given triangulation of the ‘coarse’ underlying space together with the local isotropy groups of M. We prove that, for any local system on M, this complex S(M) has the same cohomology as M. The use of S(M) in explicit calculations is illustrated in the example of the ‘teardrop’ orbifold.
The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inv... more The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors -- a natural generalization of inverse semigroups -- and extend the ESN theorem to an equivalence between this category and the category of top-heavy locally inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally inductive groupoids and to an equivalence between the category of inverse categories with oplax functors and the category of top-heavy locally inductive groupoids and ordered functors.
Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by ... more Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the development of the basic theory of latent fibrations and explores some key examples. Latent fibrations cover a wide variety of examples, some of which are partial versions of standard fibrations, and some of which are particular to partial map categories (particularly those that arise in computational settings). Latent fibrations with various special properties are identified: hyperconnected latent fibrations, in particular, are shown to support the construction of a fibrational dual; this is important to reverse differential programming and, more generally, in the theory of lenses.
Dans cet article, les auteurs etudient les categories doubles obtenues en ajoutant librement de n... more Dans cet article, les auteurs etudient les categories doubles obtenues en ajoutant librement de nouvelles cellules ou fleches a une categorie double existante. Ils discutent plus specifiquement la decidabilite de l'egalite de cellules dans la nouvelle categorie double.
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Papers by Dorette Pronk