Related papers: Protocol Choice and Iteration for the Free Corneri…
We present a term rewriting system that models the dynamic aspects of the free cornering with protocol choice of a monoidal category, which has been proposed as a categorical model of process interaction. This term rewriting system is…
We show that the category of optics in a monoidal category arises naturally from the free cornering of that category. Further, we show that the free cornering of a monoidal category is a natural setting in which to work with comb diagrams…
We present a calculus that models a simple sort of process interaction. Our calculus consists of a collection of terms together with a rewrite relation, parameterised by an arbitrary multicategory whose morphisms we understand as…
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…
Freyd categories provide a semantics for first-order effectful programming languages by capturing the two different orders of evaluation for products. We enrich Freyd categories in a duoidal category, which provides a new, third choice of…
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…
With a view on applications in computing, in particular concurrency theory and higher-dimensional rewriting, we develop notions of $n$-fold monoid and comonoid objects in $n$-fold monoidal categories and bicategories. We present a series of…
We present a unified framework for categorical systems theory which packages a collection of open systems, their interactions, and their maps into a symmetric monoidal loose right module of systems over a symmetric monoidal double category…
Interviews run on people, programs run on operating systems, voting schemes run on voters, games run on players. Each of these is an example of the abstraction pattern runs on matter. Pattern determines the decision tree that governs how a…
We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…
We present a new model of computation, described in terms of monoidal categories. It conforms the Church-Turing Thesis, and captures the same computable functions as the standard models. It provides a succinct categorical interface to most…
In the first part of this article, we give an analysis of the free monad sequence in non-cocomplete categories, with the needed colimits explicitly parametrized. This enables us to state a more finely grained functoriality principle for…
We universally characterize the produoidal category of monoidal lenses over a monoidal category. In the same way that each category induces a cofree promonoidal category of spliced arrows, each monoidal category induces a cofree produoidal…
We observe that the existence of sequential and parallel composition supermaps in higher order theories of transformations can be formalised using enriched category theory. Encouraged by relevant examples such as unitary supermaps and…
This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free…
A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for…
In this article, we show that the now classical protocol complex approach to distributed task solvability of Herlihy et al. can be understood in standard categorical terms. First, protocol complexes are functors, from chromatic (semi-)…
Applied category theory has recently developed libraries for computing with morphisms in interesting categories, while machine learning has developed ways of learning programs in interesting languages. Taking the analogy between categories…
We study rewriting for equational theories in the context of symmetric monoidal categories where there is a separable Frobenius monoid on each object. These categories, also called hypergraph categories, are increasingly relevant: Frobenius…
We build a model structure from the simple point of departure of a structured interval in a monoidal category - more generally, a structured cylinder and a structured co-cylinder in a category.