Related papers: Bilimits in categories of partial maps
This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic…
We propose the design of novel categorical generative AI architectures (GAIAs) using topos theory, a type of category that is ``set-like": a topos has all (co)limits, is Cartesian closed, and has a subobject classifier. Previous theoretical…
We explore a new connection between synthetic domain theory and Grothendieck topoi related to the distributive lattice classifier. In particular, all the axioms of synthetic domain theory (including the inductive fixed point object and the…
For a separated scheme $X$ of finite type over a perfect field $k$ of characteristic $p>0$ which admits an immersion into a proper smooth scheme over the truncated Witt ring $W_{n}$, we define the bounded derived category of locally…
Structured and decorated cospans are broadly applicable frameworks for building bicategories or double categories of open systems. We streamline and generalize these frameworks using central concepts of double category theory. We show that,…
Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite $\infty$-category theory has not been formalized. To support future work on formalizing $\infty$-category theory…
The study of weak domains and quasicontinuous domains leads to the consideration of two types generalizations of domains. In the current paper, we define the weak way-below relation between two nonempty subsets of a poset and quasiexact…
Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split…
The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category.…
The order domains are a class of commutative rings introduced by H{\o}holdt, van Lint, and Pellikaan to simplify the theory of error control codes using ideas from algebraic geometry. The definition is largely motivated by the structures…
In a Systems Engineering setting, various models are produced using a variety of methods and tools. Focusing on a type of models -- called descriptive models -- which we shall describe, we argue that, while the clarity and precision of…
We explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring $A$ using homotopy colimits. More precisely, we show that…
Here, we present a subcategory pEff of Hyland's Effective Topos Eff which can be considered a predicative variant of Eff itself. The construction of pEff is motivated by the desire of providing a "predicative" categorical universe of…
Gradually typed languages allow statically typed and dynamically typed code to interact while maintaining benefits of both styles. The key to reasoning about these mixed programs is Siek-Vitousek-Cimini-Boyland's (dynamic) gradual…
We provide, among other things: (i) a Bousfield--Kan formula for colimits in $\infty$-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) $\infty$-categorical generalizations…
Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces).…
We introduce abelian framed bicategories, which are particular framed bicategories that are locally abelian, and show that they are suitable for developing homology and cohomology theories for directed structures. This means in particular…
Topological domain walls separating 2+1 dimensional topologically ordered phases can be understood in terms of Witt equivalences between the UMTCs describing anyons in the bulk topological orders. However, this picture does not provide a…
Categories of partial functions have become increasingly important principally because of their applications in theoretical computer science. In this note we prove that the category of partial bijections between sets as an…
We show that the categories PsTop and Lim of pseudotopological spaces and limit spaces, respectively, admit cofibration category structures, and that PsTop admits a model category structure, giving several ways to simultaneously study the…