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44
Abstract and Concrete Categories. The Joy of Cats
, 2004
"... Contemporary mathematics consists of many different branches and is intimately related to various other fields. Each of these branches and fields is growing rapidly and is itself diversifying. Fortunately, however, there is a considerable amount of common ground — similar ideas, concepts, and constr ..."
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Contemporary mathematics consists of many different branches and is intimately related to various other fields. Each of these branches and fields is growing rapidly and is itself diversifying. Fortunately, however, there is a considerable amount of common ground — similar ideas, concepts, and constructions. These provide a basis for a general theory of structures.
The purpose of this book is to present the fundamental concepts and results of such a theory, expressed in the language of category theory — hence, as a particular branch of mathematics itself. It is designed to be used both as a textbook for beginners and as a reference source. Furthermore, it is aimed toward those interested in a general theory of structures, whether they be students or researchers, and also toward those interested in using such a general theory to help with organization and clarification within a special field. The only formal prerequisite for the reader is an elementary knowledge of set theory.
A universal characterization of the closed euclidean interval (Extended Abstract)
 PROC. OF 16TH ANN. IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE, LICS'01
, 2001
"... We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basi ..."
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Cited by 19 (1 self)
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We propose a notion of interval object in a category with finite products, providing a universal property for closed and bounded real line segments. The universal property gives rise to an analogue of primitive recursion for defining computable functions on the interval. We use this to define basic arithmetic operations and to verify equations between them. We test the notion in categories of interest. In the
A Dependent Type Theory with Names and Binding
 In Proceedings of the 2004 Computer Science Logic Conference, number 3210 in Lecture notes in Computer Science
, 2004
"... We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on FraenkelMostowski (FM) set theory how to address this through firstclass names: in this paper we present a dependent type theory for prog ..."
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Cited by 18 (1 self)
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We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on FraenkelMostowski (FM) set theory how to address this through firstclass names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier N , namebinding, and unique choice of fresh names. The Schanuel topos  the category underlying FM set theory  is an instance of this axiomatisation.
A Study of Categories of Algebras and Coalgebras
, 2001
"... This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and ..."
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Cited by 13 (5 self)
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This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is wellbehaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkhoff [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of "coequation" is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about "endomorphisminvariant" coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. La...
Proof theory in the abstract
 Ann. Pure Appl. Logic
, 2002
"... with great affection and respect, this small tribute to his influence. 1 Background In the Introduction to the recent text Troelstra and Schwichtenberg [44], the authors contrast structural proof theory on the one hand with interpretational proof theory on the other. They write thus. Structural proo ..."
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with great affection and respect, this small tribute to his influence. 1 Background In the Introduction to the recent text Troelstra and Schwichtenberg [44], the authors contrast structural proof theory on the one hand with interpretational proof theory on the other. They write thus. Structural proof theory is based on a combinatorial analysis of the structure of formal proofs; the central methods are cut elimination
The Biequivalence of Locally Cartesian Closed Categories and MartinLöf Type Theories
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2014
"... Seely’s paper Locally cartesian closed categories and type theory contains a wellknown result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of MartinLöf type theories with Π, Σ, and extensional identity types. However, Seely’s p ..."
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Seely’s paper Locally cartesian closed categories and type theory contains a wellknown result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of MartinLöf type theories with Π, Σ, and extensional identity types. However, Seely’s proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely’s theorem: that the BénabouHofmann interpretation of MartinLöf type theory in locally cartesian closed categories yields a biequivalence of 2categories. To facilitate the technical development we employ categories with families as a substitute for syntactic MartinLöf type theories. As a second result we prove that if we remove Πtypes the resulting categories with families with only Σ and extensional identity types are biequivalent to left exact categories.
A New Framework for Declarative Programming
, 2001
"... We propose a new indexedcategory syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over taucategories:finite product categories with canonical structure. ..."
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Cited by 10 (3 self)
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We propose a new indexedcategory syntax and semantics of Weak Hereditarily Harrop logic programming with constraints, based on resolution over taucategories:finite product categories with canonical structure.