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Articles 1 - 4 of 4
Full-Text Articles in Logic and Foundations
Category Theory And Universal Property, Niuniu Zhang
Category Theory And Universal Property, Niuniu Zhang
Honors Theses
Category theory unifies and formalizes the mathematical structure and concepts in a way that various areas of interest can be connected. For example, many have learned about the sets and its functions, the vector spaces and its linear transformation, and the group theories and its group homomorphism. Not to mention the similarity of structure in topological spaces, as the continuous function is its mapping. In sum, category theory represents the abstractions of other mathematical concepts. Hence, one could use category theory as a new language to define and simplify the existing mathematical concepts as the universal properties. The goal of …
Extending Set Functors To Generalised Metric Spaces, Adriana Balan, Alexander Kurz, Jiří Velebil
Extending Set Functors To Generalised Metric Spaces, Adriana Balan, Alexander Kurz, Jiří Velebil
Mathematics, Physics, and Computer Science Faculty Articles and Research
For a commutative quantale V, the category V-cat can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor T (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor TV on V-cat. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones.
Conceptually, this allows us to to solve the same recursive domain equation X ≅ TX in different categories (such as sets and metric spaces) and …
Constructing A Categorical Framework Of Metamathematical Comparison Between Deductive Systems Of Logic, Alex Gabriel Goodlad
Constructing A Categorical Framework Of Metamathematical Comparison Between Deductive Systems Of Logic, Alex Gabriel Goodlad
Senior Projects Spring 2016
The topic of this paper in a broad phrase is “proof theory". It tries to theorize the general
notion of “proving" something using rigorous definitions, inspired by previous less general
theories. The purpose for being this general is to eventually establish a rigorous framework
that can bridge the gap when interrelating different logical systems, particularly ones
that have not been as well defined rigorously, such as sequent calculus. Even as far as
semantics go on more formally defined logic such as classic propositional logic, concepts
like “completeness" and “soundness" between the “semantic" and the “deductive system"
is too arbitrarily defined …
Coalgebraic Logics (Dagstuhl Seminar 12411), Ernst-Erich Doberkat, Alexander Kurz
Coalgebraic Logics (Dagstuhl Seminar 12411), Ernst-Erich Doberkat, Alexander Kurz
Engineering Faculty Articles and Research
This report documents the program and the outcomes of Dagstuhl Seminar 12411 “Coalgebraic Logics”. The seminar deals with recent developments in the area of coalgebraic logic, a branch of logics which combines modal logics with coalgebraic semantics. Modal logic finds its uses when reasoning about behavioural and temporal properties of computation and communication, coalgebras have evolved into a general theory of systems. Consequently, it is natural to combine both areas for a mathematical description of system specification. Coalgebraic logics are closely related to the broader categories semantics/formal methods and verification/logic.