CyberSec Research

In the article \cite{Sim}, H. Simmons describes two monads of interests arising from the dual adjunction between the category of topological spaces and that of (bounded) distributive lattices. These are the open prime filter monad and the ideal lattice monad. It is known that the ideal lattice monad induces the ideal frame comonad on the category of frames. We show that this ideal frame comonad can be paired with the open prime filter monad via the open set-spectrum adjunction. From this, we give a new proof of the equivalence between the category of stably compact spaces and that of stably compact frames on one hand, and that of compact Hausdorff spaces and compact regular frames on the other. We show, among other things, how the \v{C}ech-Stone compactification in Pointfree Topology and Pointset Topology relate each other in this particular context.

In recent work of Lindenhovius and Zamdzhiev, it was established that the category of complete operator spaces, with completely contractive linear maps as morphisms, is locally countably presentable. In this work, we extend their conclusion to the non-complete setting and prove that the categories of operator systems, (Archimedean) order unit spaces, and unital operator algebras are all locally countably presentable as well. This is established through an analysis of forgetful functors and the identification of Eilenberg-Moore categories. We provide a complete understanding of adjunction and monadicity for forgetful functors between these categories, together with the categories of $C^*$-algebras, Banach spaces, and normed spaces. In addition, for various subcategories of function-theoretic objects, we investigate completeness and local presentability through Kadison's duality theorem.

The persistent homology transform (PHT) of a subset $M \subset \mathbb{R}^d$ is a map $\text{PHT}(M):\mathbb{S}^{d-1} \to \mathbf{Dgm}$ from the unit sphere to the space of persistence diagrams. This map assigns to each direction $v\in \mathbb{S}^{d-1}$ the persistent homology of the filtration of $M$ in direction $v$. In practice, one can only sample the map $\text{PHT}(M)$ at a finite set of directions $A \subset \mathbb{S}^{d-1}$. This suggests two natural questions: (1) Can we interpolate the PHT from this finite sample of directions to the entire sphere? If so, (2) can we prove that the resulting interpolation is close to the true PHT? In this paper we show that if we can sample the PHT at the module level, where we have information about how homology from each direction interacts, a ready-made interpolation theory due to Bubenik, de Silva, and Nanda using Kan extensions can answer both of these questions in the affirmative. A close inspection of those techniques shows that we can infer the PHT from a finite sample of heights from each direction as well. Our paper presents the first known results for approximating the PHT from finite directional and scalar data.

We introduce a notion of distributor of sites, involving suited analogs of flatness and cover-preservation, and show that this notion jointly generalizes those of morphism and comorphism of sites. Given two sites, we exhibit an adjunction between the category of distributors of sites between them and the category of geometric morphisms between the associated sheaf topoi; this adjunction restricts to an equivalence between geometric morphisms and continuous distributors of sites. We finally discuss some equipment-like properties of the bicategory of sites and distributors of sites.

We give a detailed account of the theory of enrichment over a bicategory and show that it establishes a two-fold generalization of enrichment over both quantaloids and monoidal categories. We define complete B-categories, a generalization of Cauchy-complete enriched categories serving as a basis for the development of sheaf theory in the enriched setting. We prove an adjunction between complete B-categories and 2-presheaves on the category Map(B) of left adjoints in B. We express conditions under which this adjunction becomes a left-exact reflection, yielding back the usual results linking sheaves on sites and enriched categories. We prove that our adjunction recovers the already existing results about quantaloids, and discuss the fixed points of the adjunction in the monoidal case.

We extend logical categories with fiberwise interior and closure operators so as to obtain an embedding theorem into powers of the category of topological spaces. The required axioms, besides the Kuratowski closure axioms, are a `product independence' and a `loop contraction' principle.

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of covering that we call a Galois semi-covering functor, which becomes a Galois covering when the group action is free. We study the module category of its skew group algebra under this functor. As an application, we obtain a complete description of the irreducible morphisms and almost split sequences of skew group algebras and show that the (stable) rank is preserved under skewness. In particular, we determine the stable rank of skew-gentle algebras.

We show the vanishing of higher extension groups and torsion groups between linearisation of additive functors from a semi-additive category satisfying some conditions to a category of vector spaces. In particular, we apply our results to the category of correspondences functors of Bouc-Th\'evenaz.

Given two monads $S$, $T$ on a category where idempotents split, and a weak distributive law between them, one can build a combined monad $U$. Making explicit what this monad $U$ is requires some effort. When we already have an idea what $U$ should be, we show how to recognize that $U$ is indeed the combined monad obtained from $S$ and $T$: it suffices to exhibit what we call a distributing retraction of $ST$ onto $U$. We show that distributing retractions and weak distributive laws are in one-to-one correspondence, in a 2-categorical setting. We give three applications, where $S$ is the Smyth, Hoare or Plotkin hyperspace monad, $T$ is a monad of continuous valuations, and $U$ is a monad of previsions or of forks, depending on the case. As a byproduct, this allows us to describe the algebras of monads of superlinear, resp. sublinear previsions. In the category of compact Hausdorff spaces, the Plotkin hyperspace monad is sometimes known as the Vietoris monad, the monad of probability valuations coincides with the Radon monad, and we infer that the associated combined monad is the monad of normalized forks.

We show that, when the actions of a Mazurkiewicz trace are considered not merely as atomic (i.e., mere names) but transformations from a specified type of inputs to a specified type of outputs, we obtain a novel notion of presentation for effectful categories (also known as generalised Freyd categories), a well-known algebraic structure in the semantics of side-effecting computation. Like the usual representation of traces as graphs, our notion of presentation gives rise to a graphical calculus for effectful categories. We use our presentations to give a construction of the commuting tensor product of free effectful categories, capturing the combination of systems in which the actions of each must commute with one another, while still permitting exchange of resources

We define the notion of a Lie superalgebra over a field $k$ of characteristic $2$ which unifies the two pre-existing ones - $\mathbb{Z}/2$-graded Lie algebras with a squaring map and Lie algebras in the Verlinde category ${\rm Ver}_4^+(k)$, and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect $k$), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero - the notion of a mixed Lie superalgebra over a ramified quadratic extension $R$ of the ring of Witt vectors $W(k)$.

We demonstrate that a Bousfield-Friedlander localization with a set of test morphisms in the sense introduced by Bandklayder, Bergner, Griffiths, Johnson, and Santhanam can also be characterized as a left Bousfield localization at the set of test morphisms. This viewpoint enables us to establish a homogeneous model structure associated with any calculus arising from a Bousfield-Friedlander localization of this form. As a corollary, we show that homogeneous functors in discrete calculus coincide up to homotopy with those in Goodwillie calculus. Finally, we illustrate this framework by proving that the polynomial model structure of Weiss calculus is a particular instance of tested Bousfield-Friedlander localization.

We develop a notion of covariant differential calculus for Hopf algebroids. As a byproduct, we prove analogues of the fundamental theorem of Hopf modules and a Takeuchi-Schneider equivalence in the realm of Hopf algebroids. The resulting categorical equivalences enable us to classify covariant calculi on Hopf algebroids and, more in general, covariant calculi on quantum homogeneous spaces in this context, in terms of substructures of the augmentation ideal. This generalises the well-known classification results of Woronowicz and Hermisson. A particular focus is given on examples, including covariant calculi on the Ehresmann-Schauenburg Hopf algebroid of a faithfully flat Hopf-Galois extension, and covariant calculi on scalar extension Hopf algebroids, as well as homogeneous space variants of the latter.

Using the language of moperads-monoids in the category of right modules over an operad-we reinterpret the Alekseev-Enriquez-Torossian construction of Kashiwara-Vergne (KV) solutions from associators. We show that any isomorphism between the moperad of parenthesized braids with a frozen strand and the moperad of chord diagrams gives rise to a family of genus zero KV solutions operadically generated by a single classical KV solution. We show that the Grothendieck-Teichm\"uller module groups act on the latter, intertwining the actions of the KV symmetry groups. In the other direction, we show that any symmetric KV solution gives rise to a morphism from the moperad of parenthesized braids with a frozen strand to the moperad of tangential automorphisms of free Lie algebras. This morphism factors through the moperad of chord diagrams if and only if the associated KV associator is a Drinfeld associator.

We consider the category Grpd(Asm$(A)$) of groupoids defined internally to the category of assemblies on a partial combinatory algebra $A$. In this thesis we exhibit the structure of a $\pi$-tribe on Grpd(Asm$(A)$) showing the category to be a model of type theory. We also show that Grpd(Asm$(A)$) has $W$-types with reductions and univalent object classifier for assemblies and modest assemblies, where the latter is an impredicative object classifier. Using the $W$-types with reductions, we show that Grpd(Asm$(A)$) has a model structure. Finally, we construct pGrpd(Asm$(A)$), the full subcategory of partitioned groupoid assemblies, and show that pGrpd(Asm$(A)$) has finite bilimits and bicolimits as well as showing that the homotopy category of the full subcategory of the $0$-types of pGrpd(Asm$(A)$) is RT$[A]$, the realizability topos of $A$.

We give an elementary proof of the Eilenberg-Mac Lane trace isomorphism between the third 2-abelian cohomology group and quadratic forms. Our approach yields explicit constructions and we characterize when quadratic forms can be expressed as traces of bilinear forms for arbitrary coefficient groups.

We discuss Lurie's (derived) bar and cobar constructions, the classical ones for simplicial groups and sets (due to Eilenberg-MacLane and Kan), and the classical ones for differential graded (co)algebras (due to Eilenberg-MacLane and Adams) and their relations, putting them into an abstract framework which makes sense much more generally for any cofibration of infinity-operads. Along these lines we give new and rather conceptual existence proofs of Lurie's adjunction (where bar is left adjoint) and the classical adjunction (where bar is right adjoint). We also recover various classical comparison maps, e.g. the Szczarba and Hess-Tonks maps comparing Adams cobar with Kan's loop group.

We show that in a symmetric monoidal category over a field of characteristic zero, objects with an invertible exterior power are rigid. As an application we prove some conjectures on dimensions in symmetric monoidal categories recently stated by Baez, Moeller and Trimble.