Related papers: Axiomatic Method and Category Theory
A new technique is proposed to classify a topological field in abelian lattice gauge theories. We perform the classification by regarding the topological field as a local composite field of the gauge field tensor instead of the vector…
We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a…
The class of generic structures among those consisting of the measure algebra of a probability space equipped with an automorphism is axiomatizable by positive sentences interpreted using an approximate semantics. The separable generic…
Here, by introducing a version of "Unexpected hanging paradox" we try to open a new way and a new explanation for paradoxes, similar to liar paradox. Also, we will show that we have a semantic situation which no syntactical logical system…
This paper investigates Voevodsky's univalence axiom in intensional Martin-L\"of type theory. In particular, it looks at how univalence can be derived from simpler axioms. We first present some existing work, collected together from various…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We give a model-theoretic characterization of the class of geometric theories classified by an atomic topos having enough points; in particular, we show that every complete geometric theory classified by an atomic topos is countably…
Despite recent advances in automating theorem proving in full first-order theories, inductive reasoning still poses a serious challenge to state-of-the-art theorem provers. The reason for that is that in first-order logic induction requires…
Grothendieck toposes, and by extension, logical theories, can be represented by topological structures. Butz and Moerdijk showed that every topos with enough points can be represented as the topos of sheaves on an open topological groupoid.…
The semantics of extensional type theory has an elegant categorical description: models of extensional =-types, 1-types, and Sigma-types are biequivalent to finitely complete categories, while adding Pi-types yields locally Cartesian closed…
The term ``Boolean category'' should be used for describing an object that is to categories what a Boolean algebra is to posets. More specifically, a Boolean category should provide the abstract algebraic structure underlying the proofs in…
The theory of associative $n$-categories has recently been proposed as a strictly associative and unital approach to higher category theory. As a foundation for a proof assistant, this is potentially attractive, since it has the potential…
We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that…
We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and…
This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and Blechschmidt. The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras…
Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson, represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector in his vision of a homotopy…
We formulate and discuss a general axiomatic theory of arbitrary objects. This theory is expressed in a simple first-order language without modal operators, and it is governed by classical logic.
We develop an analogue of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov's method of axiomatization of probability theory: main distinguishing features of frequency probabilities are taken…
The following offers a new axiomatic basis of mechanics and physics in their most important dynamics domain, i. e. an axiom (principle) of completeness intended to generalize Newton's second law of motion for the case of a non-stationary…
We introduce some classes of genuine higher categories in homotopy type theory, defined as well-behaved subcategories of the category of types. We give several examples, and some techniques for showing other things are not examples. While…