arXiv subjects

Kevin Carlson

Publications and source records attributed to Kevin Carlson.

7 recordsLinked to original sources

arXiv2026

Comonads as spaces

Comonads on Set generalize both categories and topological spaces. Expanding upon Garner's work on ionads, we develop aspects of the theory of topological spaces for arbitrary comonads on arbitrary categories. Our approach is centered around density comonads, which provide an abstraction of subbases. We study subbases as well as bases in terms of density comonads, and we study continuous maps of comonads in terms of functors between coalgebra categories, with definitions that recover the usual notions for topological spaces. Whereas Ahman and Uustalu characterized categories as precisely the polynomial comonads on Set, we characterize topological spaces as precisely the density comonads of diagrams of subsets of a set, which are familiar as topological subbases. We show that every comonad on Set has an underlying topological space, and that this construction is a reflection with respect to continuous maps; similarly, every comonad on Set has an underlying small category, and this construction is a coreflection. We also show that the category of all comonads on Set with continuous maps is complete, and that its full subcategory of accessible comonads is cocomplete. Continuous maps and ordinary comonad morphisms form a double category, which, in the case of the polynomial comonads on Set, recovers the double category of functors and retrofunctors of Clarke and Di Meglio. We find topological intuition for these concepts in terms of "halos", an abstraction of infinitesimal neighborhoods of points, defined as formal limits of neighborhood systems. We include a long appendix of counterexamples, many applicable to general (co)monad theory rather than the particular concerns of this text.

math.CT↗

arXiv2026

A Categorical Approach to Semantic Interoperability across Building Lifecycle

Buildings generate heterogeneous data across their lifecycle, yet integrating these data remains a critical unsolved challenge. Despite three decades of standardization efforts, over 40 metadata schemas now span the building lifecycle, with fragmentation accelerating rather than resolving. Current approaches rely on point-to-point mappings that scale quadratically with the number of schemas, or universal ontologies that become unwieldy monoliths. The fundamental gap is the absence of mathematical foundations for structure-preserving transformations across heterogeneous building data. Here we show that category theory provides these foundations, enabling systematic data integration with $O(n)$ specification complexity for $n$ ontologies. We formalize building ontologies as first-order theories and demonstrate two proof-of-concept implementations in Categorical Query Language (CQL): 1) generating BRICK models from IFC design data at commissioning, and 2) three-way integration of IFC, BRICK, and RealEstateCore where only two explicit mappings yield the third automatically through categorical composition. Our correct-by-construction approach treats property sets as first-class schema entities and provides automated bidirectional migrations, and enables cross-ontology queries. These results establish feasibility of categorical methods for building data integration and suggest a path toward an app ecosystem for buildings, where mathematical foundations enable reliable component integration analogous to smartphone platforms.

cs.DB↗

arXiv2025

Comparing loose bimodules and double barrels using pseudo-models of enhanced sketches

(Pseudo) double categories have two sorts of morphisms: tight ones which compose strictly, and loose ones which compose up to coherent isomorphism. In this paper, we consider bimodules between double categories in the loose direction. We provide two formulation of this concept -- first as pseudo-bimodules between pseudo-categories in the 2-category of categories, and second as double barrels generalizing Joyal's definition of bimodules between categories as functors into the walking arrow -- and prove these two formulations equivalent. In order to prove this equivalence, we define a notion of \emph{pseudo-model} of an enhanced sketch, which may be of independent interest. We then consider some double category theory unlocked by the theory of loose bimodules: loose adjunctions, and loose limits.

math.CT↗

arXiv2025

Presheaves on lax double functors; or, Instances of models of double theories

We introduce a notion of (co)presheaf on a lax double functor $X$, which we generally call an instance. In the terminology of double-categorical logic, a lax double functor valued in sets, possibly preserving finite products, is called a model of a double (Lawvere) theory. By varying the double theory, we uniformly define a well-behaved notion of instances of categories, profunctors, monads, monoidal categories, multicategories, and more, and we recover for instance the multifunctors into the category of sets in the last example. We show that instances of $X$ can be described either in terms of modules from the terminal model $I$ to $X,$ satisfying an additional condition on triviality of the left action, or as loose natural transformations from $I$ to $X.$ We propose a notion of discrete opfibration between models of a double theory, establish a comprehensive factorization system, and prove an elements correspondence giving an equivalence between the category of instances of and the category of discrete opfibrations over a model $X.$ We describe properties of the resulting categories of instances, relying on a "collage" construction which we characterize as a lax colimit of a model of a double theory. An appendix gives a detailed treatment of certain morphisms of lax functors relevant also for bicategory theory: (loose) transformations versus modules and modifications versus modulations.

math.CT↗

arXiv2025

Porous Convection in the Discrete Exterior Calculus with Geometric Multigrid

The discrete exterior calculus (DEC) defines a family of discretized differential operators which preserve certain desirable properties from the exterior calculus. We formulate and solve the porous convection equations in the DEC via the Decapodes.jl embedded domain-specific language (eDSL) for multiphysics problems discretized via CombinatorialSpaces.jl. CombinatorialSpaces.jl is an open-source Julia library which implements the DEC over simplicial complexes, and now offers a geometric multigrid solver over maps between subdivided simplicial complexes. We demonstrate numerical results of multigrid solvers for the Poisson problem and porous convection problem, both as a standalone solver and as a preconditioner for open-source Julia iterative methods libraries.

cs.CE↗

arXiv2024

Flight Demonstration and Model Validation of a Prototype Variable-Altitude Venus Aerobot

This paper details a significant milestone towards maturing a buoyant aerial robotic platform, or aerobot, for flight in the Venus clouds. We describe two flights of our subscale altitude-controlled aerobot, fabricated from the materials necessary to survive Venus conditions. During these flights over the Nevada Black Rock desert, the prototype flew at the identical atmospheric densities as 54 to 55 km cloud layer altitudes on Venus. We further describe a first-principle aerobot dynamics model which we validate against the Nevada flight data and subsequently employ to predict the performance of future aerobots on Venus. The aerobot discussed in this paper is under JPL and Aerostar development for an in-situ mission flying multiple circumnavigations of Venus, sampling the chemical and physical properties of the planet's atmosphere and also remotely sensing surface properties.

cs.RO↗

arXiv2012

Omitting types and AF algebras

We give model-theoretic characterizations of UHF algebras and of AF algebras as C*-algebras that omit certain sets of types.

math.LO↗