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.

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 Eckmann-Hilton 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 two-fold monoidal categories (with weak units) are degenerate symmetric. In particular, strictly associative oneobject, one-arrow 3-groupoids (with weak units) cannot realise all simply-connected homotopy 3-types. 1. Introduction and

– Dr. rer. nat. –

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 Yang-Baxter 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)

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 Yang-Baxter 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)

Note on commutativity in double semigroups and two-fold monoidal categories

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

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 p-form electrodynamics, in which (p − 1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups 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 non-Abelian symmetry groups. The main new feature is that our model involves both parallel transports along paths and generalized transports along surfaces with a non-trivial 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 non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian 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.