Characterizations of Morita equivalent inverse semigroups

We prove that four different notions of Morita equivalence for inverse semigroups motivated by C^∗-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the category of unitary actions of an inverse semigroup is monadic over the category of étale actions. Consequently, the category of unitary actions of an inverse semigroup is equivalent to the category of presheaves on its Cauchy completion. More generally, we prove that the same is true for the category of closed actions, which is used to define the Morita theory in semigroup theory, of any semigroup with right local units.     

The fundamental progroupoid of a general topos

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and cannot be replaced by a localic groupoid. The classifying topos is no longer a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver, July 2004.     

Structures de dérivabilité

We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of “homotopical categories,” and by Radulescu-Banu in the context of Cisinski's “derivable categories.”     

Cosheaves and connectedness in formal topology

The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π₀ together with certain additional structure and properties that constrain π₀ to be the connected components cosheaf. In the inductively generated case, complete spreads (in the sense of Bunge and Funk) corresponding to cosheaves are defined as formal topologies. Maps between the complete spreads are equivalent to homomorphisms between the cosheaves. A cosheaf is the connected components cosheaf for a locally connected formal topology iff its complete spread is a homeomorphism, and in this case it is a terminal cosheaf.A new, geometric proof is given of the topos-theoretic result that a cosheaf is a connected components cosheaf iff it is a “strongly terminal” point of the symmetric topos, in the sense that it is terminal amongst all the generalized points of the symmetric topos. It is conjectured that a study of sites as “formal toposes” would allow such geometric proofs to be incorporated into predicative mathematics.     

An “almost” full embedding of the category of graphs into the category of groups

We construct a functor F:Graphs→Groups which is faithful and “almost” full, in the sense that every nontrivial group homomorphism FX→FY is a composition of an inner automorphism of FY and a homomorphism of the form Ff, for a unique map of graphs f:X→Y. When F is composed with the Eilenberg-Mac Lane space construction K(FX,1) we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null-homotopic maps.We provide several applications of this construction to localizations (i.e. idempotent functors); we show that the questions:

(1)

Is every orthogonality class reflective?

(2)

Is every orthogonality class a small-orthogonality class?

have the same answers in the category of groups as in the category of graphs. In other words they depend on set theory: (1) is equivalent to weak Vopěnka's principle and (2) to Vopěnka's principle. Additionally, the second question, considered in the homotopy category, is also equivalent to Vopěnka's principle.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

Protomodular aspect of the dual of a topos

The structure of the dual of a topos is investigated under its aspect of a Barr exact and protomodular category. In particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized, and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms.     

Formal plethories

Unstable operations in a generalized cohomology theory E   give rise to a functor from the category of algebras over _E?$E_{?}$ to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where _E?$E_{?}$ is a Prüfer ring. I show that the “logarithmic” functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.     

Injective power objects and the axiom of choice

“The axiom of choice states that any set X of non-empty sets has a choice function—i.e. a function satisfying f(x)∈x for all x∈X. When we want to generalise this to a topos, we have to choose what we mean by non-empty, since in , the three concepts non-empty, inhabited, and injective are equivalent, so the axiom of choice can be thought of as any of the three statements made by replacing “non-empty” by one of these notions.It seems unnatural to use non-empty in an intuitionistic context, so the first interpretation to be used in topos theory was the notion based on inhabited objects. However, Diaconescu (1975) [1] showed that this interpretation implied the law of the excluded middle, and that without the law of the excluded middle, even the finite version of the axiom of choice does not hold! Nevertheless some people still view this as the most appropriate formulation of the axiom of choice in a topos.In this paper, we study the formulation based upon injective objects. We argue that it can be considered a more natural formulation of the axiom of choice in a topos, and that it does not have the undesirable consequences of the inhabited formulation. We show that if it holds for , then it holds in a wide variety of topoi, including all localic topoi. It also has some of the classical consequences of the axiom of choice, although a lot of classical results rely on both the axiom of choice and the law of the excluded middle. An additional advantage of this formulation is that it can be defined for a slightly more general class of categories than just topoi.We also examine the corresponding injective formulations of Zorn’s lemma and the well-order principle. The injective form of Zorn’s lemma is equivalent to the axiom of injective choice, and the injective well-order principle implies the axiom of injective choice.     

A remark on K-theory and S-categories

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories? ”     

Incidence categories

Given a family F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category C_F called the incidence category ofF. This category is “nearly abelian” in the sense that all morphisms have kernels/cokernels, and possesses a symmetric monoidal structure akin to direct sum. The Ringel-Hall algebra of C_F is isomorphic to the incidence Hopf algebra of the collection P(F) of order ideals of posets in F. This construction generalizes the categories introduced by K. Kremnizer and the author, in the case when F is the collection of posets coming from rooted forests or Feynman graphs.     

Quadratic functors on pointed categories

We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups Ab, and whose source category is an arbitrary category C with null object such that all objects are colimits of copies of a generating object E which is small and regular projective; this includes all categories of models V of a pointed theory T. More specifically, we are interested in such quadratic functors F from C to Ab which preserve filtered colimits and suitable coequalizers.A functorial equivalence is established between such functors F:C→Ab and certain minimal algebraic data which we call quadratic C-modules: these involve the values on E of the cross-effects of F and certain structure maps generalizing the second Hopf invariant and the Whitehead product.Applying this general result to the case where E is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for C being the category of groups or of modules over some ring; here quadratic C-modules are equivalent with abelian square groups or quadratic R-modules, respectively.     

The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra (Theory Appl Categ 24(7):117–147, 2010). For the topos of sets, we show that torsion-free functors on Loganathan’s category L(S) of an inverse semigroup S are equivalent to a special class of non-strict representations of S, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S. We describe the correspondence between directed and pullback preserving functors on L(S) and transitive and effective representations of S, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an S-torsor, of an inverse semigroup S in the topos of sheaves \({\mathsf {Sh}}(X)\) on a topological space X. We prove that the category of filtered functors from L(S) to the topos \({\mathsf {Sh}}(X)\) is equivalent to the category of universal representations of S in \({\mathsf {Sh}}(X)\). We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on L(S).     

Monoreflections of Archimedean ?-groups, regular σ-frames and regular Lindelöf frames

We prove that in the category of Archimedean lattice-ordered groups with weak unit there is no homomorphism-closed monoreflection strictly between the strongest essential monoreflection (the so-called “closure under countable composition”) and the strongest monoreflection (the epicompletion). It follows that in the category of regular σ-frames, the only non-trivial monoreflective subcategory that is hereditary with respect to closed quotients consists of the boolean σ-algebras. Also, in the category of regular Lindelöf locales, there is only one non-trivial closed-hereditary epi-coreflection. The proof hinges on an elementary lemma about the kinds of discontinuities that are exhibited by the elements of a composition-closed l-group of real-valued functions on R.     

Commutator theory, action groupoids, and an intrinsic Schreier-Mac Lane extension theorem

Beyond groups of automorphisms in the category Gp of groups and Lie-algebras of derivations in the category K-Lie of Lie algebras, there are structures of internal groupoids (called action groupoids) in both categories. They allow a synthesis of the notion of obstruction to extensions. This leads, in any pointed protomodular category C with split extension classifiers, to a general treatment of non-abelian extensions which can be understood as morphisms in a certain groupoid TorsC.     

The Isbell monad

In 1966 [7], 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 [5] from orthogonal classes of arrows to orthogonal classes of cocones and cones.     

A remark on Mackey-functors

In the following note we characterize the category of Mackey functors from a categoryC, satisfying a few assumptions, to a categoryD as the category of functors from Sp(C), the category of spans inC, toD which preserve finite products. This caracterization permits to apply all results on categories of functors preserving a given class of limits to the case of Mackey-functors.     

Cohomologically trivial internal categories in categories of groups with operations

We describe cohomologically trivial internal categories in the categoryC of groups with operations satisfying certain conditions ([15], [16]). As particular cases we obtain: ifC=Gr, H⁰(C, –)=0 iff C is a connected internal category; ifC=Ab,H ¹(C, –)=0 iff C is equivalent to the discrete internal category (Cokerd, Cokerd, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the categoryC.     

Descent in ∗ -autonomous categories

We extend the result of Joyal and Tierney asserting that a morphism of commutative algebras in the ∗-autonomous category of sup-lattices is an effective descent morphism for modules if and only if it is pure, to an arbitrary ∗-autonomous category V (in which the tensor unit is projective) by showing that any V-functor out of V is precomonadic if and only if it is comonadic.     

Lifted generalized permutahedra and composition polynomials

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a “lifting” construction for these polytopes, which turns an n  -dimensional generalized permutahedron into an (n+1)$(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff ?s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra”, answering two questions of Devadoss and Forcey.