Hypergraph Categories

  title={Hypergraph Categories},
  author={Brendan Fong and David I. Spivak},
Supplying bells and whistles in symmetric monoidal categories.
It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in
Structured Cospans
One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and
Regular and relational categories: Revisiting 'Cartesian bicategories I'
Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of
Monoidal Grothendieck Construction
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the
String Diagrams for Regular Logic (Extended Abstract)
The syntax and proof rules of regular logic are understood in terms of the free regular category FRg(T) on a set T to show that every natural category has an associated regular calculus, and conversely from every regular calculus one can construct a regular category.
C T ] 2 0 Ju n 20 19 Graphical Regular Logic
Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof
String Diagram Rewrite Theory I: Rewriting with Frobenius Structure
This work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting, and proves this interpretation to be sound and complete and shows that the approach can be generalised to rewrite modulo multiple Frobenii structures.
Wiring diagrams as normal forms for computing in symmetric monoidal categories
An "unbiased" approach to implementing symmetric monoidal categories, based on an operad of directed, acyclic wiring diagrams, is presented, because the interchange law and other laws of a SMC hold identically in a wiring diagram.
Regular Calculi I: Graphical Regular Logic
What is ergonomic syntax for relations? In this first paper in a series of two, to answer the question we define regular calculi: a suitably structured functor from a category representing the syntax


Decorated Cospans
Let C be a category with finite colimits, writing its coproduct +, and let (D,⊗) be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided
The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits
It is shown that wiring diagrams form the morphisms of an operad $\mcT$, capturing this self-similarity, and is moved on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well.
Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams
This paper uses the language of operads to study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones, and defines two W-algebras, G and L, which associate semantic content to the structures in W.
AbstractTHE calculus of matrices has had a curious history. It was first used by Hamilton in 1853 under the name of “Linear and Vector Functions”. Cay ley used the term matrix in 1854, and developed
We show that the generic symmetric monoidal category with a commu- tative separable algebra which has a Σ-family of actions is the category of cospans of finite Σ-labelled graphs restricted to finite
A compositional framework for Markov processes
It is proved that black boxing gives a symmetric monoidal dagger functor sending open detailed balanced Markov processes to open circuits made of linear resistors, and described how to “black box” an open Markov process.
A Compositional Framework for Passive Linear Networks
Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where
The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes
This work investigates the hierarchical structure of processes using the mathematical theory of operads, and defines an operad of black boxes and directed wiring diagrams of processes (which it is called propagators, after Radul and Sussman), which are useful for modeling dynamic flows of information.