Related papers: Towards a double operadic theory of systems
A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can…
In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold…
Operads may be represented as symmetric monoidal functors on a small symmetric monoidal category. We discuss the axioms which must be imposed on a symmetric monoidal functor in order that it give rise to a theory similar to the theory of…
This paper studies the existence of model category structures on algebras and modules over operads in monoidal model categories.
In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a \emph{network model}, a monoidal variant of…
Dynamical systems---by which we mean machines that take time-varying input, change their state, and produce output---can be wired together to form more complex systems. Previous work has shown how to allow collections of machines to…
In this chapter a general mathematical framework for probabilistic theories of operationally understood circuits is laid out. Circuits are comprised of operations and wires. An operation is one use of an apparatus and a wire is a…
We extend the free cornering of a symmetric monoidal category, a double categorical model of concurrent interaction, to support branching communication protocols and iterated communication protocols. We validate our constructions by showing…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure.…
We build on the correspondence between Petri nets and free symmetric strict monoidal categories already investigated in the literature, and present a categorical semantics for Petri nets with guards. This comes in two flavors: Deterministic…
Applications of category theory often involve symmetric monoidal categories (SMCs), in which abstract processes or operations can be composed in series and parallel. However, in 2020 there remains a dearth of computational tools for working…
Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss…
Fong developed `decorated cospans' to model various kinds of open systems: that is, systems with inputs and outputs. In this framework, open systems are seen as the morphisms of a category and can be composed as such, allowing larger open…
This paper presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category…
We propose a categorical framework for processes which interact bidirectionally with both an environment and a 'controller'. Examples include open learners, in which the controller is an optimiser such as gradient descent, and an approach…
We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty…
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…
We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…