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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “many-topoi” view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a

Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences

. In this contribution to the structure theory of semigroups, we propose a unified generalisation of a string of results on group extensions, originating on the one hand in the seminal structure and covering theorems of McAlister and on the other, in Ash's celebrated solution of the Rhodes conjecture in finite semigroup theory. McAlister proved that each inverse monoid admits an E-unitary cover, and gave a structure theorem for E-unitary inverse monoids. Subsequent generalisations extended one or both results to orthodox monoids (McAlister, Szendrei, Takizawa), regular monoids (Trotter), E-dense semigroups in which the idempotents form a semilattice (Margolis and Pin, Fountain), and E-dense semigroups in which the idempotents form a subsemigroup (Almeida, Pin and Weil, Zhonghao Jiang). We show that any E-dense monoid admits a D-unitary E-dense cover and we provide a structure theorem for D-unitary E-dense monoids, in terms of groups acting on a category. Here D(M) is the least weakly s...

This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists.

Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modal-structuralism and a category theoretic approach as remaining non-absolutist

This paper shows that OO principles can be used to enhance the rigour of mathematical notation without loss of brevity and clarity. It is well known that traditional mathematical notation is not completely formal. This is so because mathematicians and other users of mathematical notation tend to sacrifice exactness to obtain brevity and clarity. The mathematician thereby leaves to the reader to guess the meaning of each formula presented based on the written and unwritten rules of the particular field of research. This works perfectly well for communication between researchers in the same field, but may be an obstacle for communication between researchers form different fields or for newcomers such as students. The lack of rigour in mathematical notation may also be an obstacle when mathematical phenomena are to be simulated on computers, where the programmer has to fill out the gaps in the notation. It is generally believed that complete formal rigour leads to an explosio...

syntax of elementary languages in Gedanken . . . . . . . . . . . 129 7.1 Match counting algorithm for patterns over PC k . . . . . . . . . . . . . 157 8.1 log 2 speed of Model Graphs after elimination . . . . . . . . . . . . . . . 187 8.2 log 2 speed-up of Model Graphs after elimination . . . . . . . . . . . . . 188 8.3 log 2 speed of Model Graphs after invalidation . . . . . . . . . . . . . . . 188 8.4 log 2 speed-up of Model Graphs after invalidation . . . . . . . . . . . . . 189 ix Chapter 1 Summary One goal of computer science has been to develop a tool T to aid a programmer in building a program P that satisfies a specification S by helping the programmer build a proof in some logic of programs L that shows that P satisfies S. S typically is a pair of propositions (#, #) such that, for an input x to P , #(x) # #(P (x)) when P is defined on x. # is called the precondition or assumption, and # is called the postcondition or assertion. The problem of finding a suitable logic L of programs and specifications and verification tool T may be generically referred to as the "Floyd-Hoare problem", formulated around 1967 [Flo67, Hoa69]. Around 1977, Davis and Schwartz proposed an extension of the Floyd-Hoare problem in which there are multiple assumptions and assertions, referring to the state of a program as execution passes through di#erent places # in the program [DS77, Sch77]. A placed proposition is then a pair (#, #), where # is either a line of a program or the name of a function. A placed proposition (#, #) holds when, if execution reaches # and the value of the variables X in P is V , then #(V ) is valid. A program with assumptions and assertions or praa is then a triple R = (P, E, F ) where the assumptions E and assertions F are sets of placed propositions. T...

Summary. We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the terminal object is a nontrivial indecomposable projective. The incompleteness theorem showed that, in the classical case, it is not enough to look only at those local toposes in which all the numerals are standard. Thus, for a classical mathematician, Hilbert’s formalist program is not compatible with the belief in a Platonic standard model. However, for pure intuitionistic type theory, a single model suffices, the linguistically constructed free topos, which is the initial object in the category of all elementary toposes and logical functors. Hence, for a moderate intuitionist, formalism and Platonism can be reconciled after all. The completeness theorem can be sharpened to represent any topos by continuous sections of a sheaf of local toposes. 1.1

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