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Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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Cited by 74 (6 self)
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
Adjointness in foundations
 Dialectica
, 1969
"... Author’s commentary In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globali ..."
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Cited by 22 (0 self)
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Author’s commentary In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globalized into functors, such as Spec, carrying much more information. Also, “theories ” (even when presented symbolically) are viewed explicitly as categories; so are the background universes of sets that serve as the recipients for models. (Models themselves are functors, hence preserve the fundamental operation of substitution/composition in terms of which the other logical operations can be characterized as local adjoints.) My 1963 observation (referred to by Eilenberg and Kelly in La Jolla, 1965), that cartesian closed categories serve as a common abstraction of type theory and propositional logic, permits an invariant algebraic treatment of the essential problem of proof theory, though most of the later work by proof theorists still relies on presentationdependent formulations. This article sums up a stage of the development of the relationship between
An introduction to ncategories
 In 7th Conference on Category Theory and Computer Science
, 1997
"... ..."
Pasting Schemes for the Monoidal Biclosed Structure on
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a gen ..."
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Cited by 18 (0 self)
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this. Contents 1 Introduction 3 2 Cubes and cubical sets 5 2.1 Cubes combinatorially : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 A model category for cubes : : : : : : : : : : : : : : : : : : : : : 6 2.3 Generating the model category for cubes : : : : : : : : : : : : : : 7 2.4 Cubical sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Duality : : : : : : : : : : : : : ...
Quantaloids describing causation and propagation for physical properties
 Foundations of Physics Letters
, 2001
"... We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems. ..."
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Cited by 17 (9 self)
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We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems.
Taking categories seriously
 Revista Colombiana de Matem'aticas
, 1986
"... Such a universal instrument for guiding the learning, development, and use of advanced mathematics does not fail to have its indications also in areas of school and college mathematics, in the most basic relationships of space and quantity and the calculations based on those relationships. In saying ..."
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Cited by 14 (0 self)
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Such a universal instrument for guiding the learning, development, and use of advanced mathematics does not fail to have its indications also in areas of school and college mathematics, in the most basic relationships of space and quantity and the calculations based on those relationships. In saying "take categories seriously", I advocate noticing, cultivating,
Introducing categories to the practicing physicist. In: What is Category Theory
 Advanced Studies in Mathematics and Logic 30, pp.45–74, Polimetrica Publishing
, 2006
"... We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra o ..."
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Cited by 12 (7 self)
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We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra of practicing physics. We will not provide rigorous definitions or anything resembling a coherent mathematical theory, but we will take the reader for a journey introducing concepts which are part of category theory in a manner that the physicist will recognize them. 1 Why? Why would a physicist care about category theory, why would he want to know about it, why would he want to show off with it? There could be many reasons. For example, you might find John Baez’s webside one of the coolest in the world. Or you might be fascinated by Chris Isham’s and Lee Smolin’s ideas on the use of topos theory in Quantum Gravity. Also the connections between knot theory, braided categories, and sophisticated mathematical physics such as quantum groups and topological quantum field theory might lure you. Or, if you are also into pure mathematics, you might just appreciate category theory due to its incredible unifying power of mathematical structures and constructions. But there is a far more onthenose reason which is never mentioned. Namely, a category is the exact mathematical structure of practicing physics! What do I mean here by a practicing physics? Consider a physical system of type A (e.g. a qubit, or two qubits, or an electron, or classical measurement data) and perform an operation f on it (e.g. perform a measurement on it) which results in a system possibly of a different type B (e.g. the system together with classical data which encodes the measurement outcome, or, just classical data in the case that the measurement destroyed the system). So typically we have
Kindergarten quantum mechanics — lecture notes
 In: Quantum Theory: Reconsiderations of the Foundations III
, 2005
"... Abstract. These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substanti ..."
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Cited by 9 (7 self)
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Abstract. These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of socalled Strongly Compact Closed Categories (introduced by Abramsky and I in [3, 4]) which subsumes my Logic of Entanglement [11]. For a survey on the ‘what’, the ‘why ’ and the ‘hows ’ I refer to a previous set of lecture notes [12, 13]. In a last section we provide some pointers to the body of technical literature on the subject.
On the Role of Category Theory in the Area of Algebraic Specifications
 In LNCS , Proc. WADT11
, 1996
"... . The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing pa ..."
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Cited by 9 (2 self)
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. The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of `classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has influenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review t...