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The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
, 2007
"... Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained ..."
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Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.
Algebraic foundations for effectdependent optimisations
 In POPL
, 2012
"... We present a general theory of Giffordstyle type and effect annotations, where effect annotations are sets of effects. Generality is achieved by recourse to the theory of algebraic effects, a development of Moggi’s monadic theory of computational effects that emphasises the operations causing the e ..."
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Cited by 6 (1 self)
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We present a general theory of Giffordstyle type and effect annotations, where effect annotations are sets of effects. Generality is achieved by recourse to the theory of algebraic effects, a development of Moggi’s monadic theory of computational effects that emphasises the operations causing the effects at hand and their equational theory. The key observation is that annotation effects can be identified with operation symbols. We develop an annotated version of Levy’s CallbyPushValue language with a kind of computations for every effect set; it can be thought of as a sequential, annotated intermediate language. We develop a range of validated optimisations (i.e., equivalences), generalising many existing ones and adding new ones. We classify these optimisations as structural, algebraic, or abstract: structural optimisations always hold; algebraic ones depend on the effect theory at hand; and abstract ones depend on the global nature of that theory (we give modularlycheckable sufficient conditions for their validity).
NOTE ON COMMUTATIVITY IN DOUBLE SEMIGROUPS AND TWOFOLD MONOIDAL CATEGORIES
"... A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and a ..."
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A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but it does not use units. The result implies in particular that all cancellative double semigroups and all inverse double semigroups are commutative. Stepping up one dimension, the result is used to prove that all strictly associative twofold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative oneobject, onearrow 3groupoids (with weak units) cannot realise all simplyconnected homotopy 3types. 1. Introduction and
Medial Commutativity
, 2007
"... It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), called medial commutativity. Medial commutativity in the presence of the unit object enables us to define assoc ..."
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It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor ∧. This preservation boils down to an isomorphic representation of the YangBaxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor ∧. In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups S n for 0 ≤ i ≤ n. i)
unknown title
, 2006
"... It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), which we propose to call commixing. Commixing in the presence of the unit object enables us to define associati ..."
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It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), which we propose to call commixing. Commixing in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of commixing by the biendofunctor ∧. This preservation boils down to an isomorphic representation of the YangBaxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor ∧. In the final section one finds coherence conditions for commixing in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups S n for 0 ≤ i ≤ n. i)
unknown title
, 2006
"... Note on commutativity in double semigroups and twofold monoidal categories ..."
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Note on commutativity in double semigroups and twofold monoidal categories
Bicartesian Coherence
, 2007
"... Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free ..."
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Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free
DAMTP200327 Higher Gauge Theory and a nonAbelian generalization
, 2003
"... In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of ..."
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In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G = U(1), there exists a generalization, known as pform electrodynamics, in which (p − 1)dimensional charged objects can be propagated along psurfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed psurfaces. In this article, we use Lie 2groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p = 2 to possibly nonAbelian symmetry groups. The main new feature is that our model involves both parallel transports along paths and generalized transports along surfaces with a nontrivial interplay of these two types of variables. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be nonAbelian and which others are always Abelian. A discrete version of connections on nonAbelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.