Received Milner's action calculus implements abstraction in monoidal categories, so that familiar-calculi can be subsumed together with the-calculus and the Petri nets. Variables are generalised to names: only a restricted form of substitution is allowed. In the present paper, the well-known categorical semantics of the-calculus is generalised to the action calculus. A suitable functional compl...

Introduction The development of the formal theory of monads, begun in [23] and continued in [15], shows that much of the theory of monads [1] can be generalized from the setting of the 2-category Cat of small categories, functors and natural transformations to that of a general 2-category. The generalization, which involves defining the 2-category Mnd(K) of monads, monad maps and monad 2-cells ...

Abstract Given a marked $$\infty $$ ∞ -category $$\mathcal {D}^{\dagger }$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">D† (i.e. an equipped with specified collection of morphisms) and functor $$F: \mathcal {D}\rightarrow ...

We develop rigorous foundations for parametrized homotopy theory. After preliminaries on point-set topology, base change functors, and proper actions of non-compact Lie groups, we develop the homotopy theory of equivariant ex-spaces (spaces over and under a given space) and of equivariant parametrized spectra. We emphasize several issues of independent interest and include a good deal of new ma...

We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of pro-monoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this – and the simplest, once the (substantial...

Nick Gurski’s new book addresses some central concerns of the subject known as higher category theory; and yet, it is some distance from what many people now understand by that term. This may puzzle some. I will therefore begin by locating Gurski’s book within the mathematical landscape. Let n ∈ N ∪ {∞}. Roughly, an n-category consists of some objects, some 1-morphisms between objects, some 2-m...

Relational languages such as Ruby are used to derive hardware circuits from abstract speci cations of their behaviour. Much reasoning is done informally in Ruby using pictorial representations of relational terms. We formalise this use of pictures in circuit design. We show that pictures naturally form a unitary pretabular allegory. Homomorphisms of pictures correspond to adding new wires or ci...

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection fr...

We define weak units in a semi-monoidal 2-category C as cancellable pseudo-idempotents: they are pairs (I, α) where I is an object such that tensoring with I from either side constitutes a biequivalence of C , and α : I ⊗ I → I is an equivalence in C . We show that this notion of weak unit has coherence built in: Theorem A: α has a canonical associator 2-cell, which automatically satisfies the ...

namely 90 a M0 a M! a M1 a Yset : set !Mset: We recall from [8] or [9] that a locally small category B is said to be total (abbreviating totally cocomplete) if Y : B !MB has a left adjoint, X: Considerable motivation for the terminology is given in either reference. Examples include categories of algebras, categories of spaces and categories of sheaves on a Grothendieck site. The reader is advi...